The Apery Limits for all summands of the form, k binomial(n, k) binomial(a1 n + b1 k, k) binomial(a2 n + b2 k, k) x , for a1 and a2 from 1 to, 2, and b1 and b2 from 0 to, 2, and x from to, 3 By Shalosh B. Ekhad ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 3 ) binomial(n, k) / ----- k By Shalosh B. Ekhad Let ----- \ 3 b(n) = ) binomial(n, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 2 2 (7 n - 7 n + 2) b(n - 1) 8 (n - 1) b(n - 2) b(n) = ------------------------- + ------------------- 2 2 n n and in Maple notation b(n) = (7*n^2-7*n+2)/n^2*b(n-1)+8*(n-1)^2/n^2*b(n-2) Of course, the initial conditions are b(0) = 1, b(1) = 2 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 2 2 (7 n - 7 n + 2) a(n - 1) 8 (n - 1) a(n - 2) a(n) = ------------------------- + ------------------- 2 2 n n but with the following simpler initial conditions a(0) = 0, a(1) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.41123351671205660911810379166150629730473747530169960943388955734250186760\ 08002184584072251549396763 2 Pi This constant is identified as, --- 24 The implied delta is, -0.4851864691142866517201640765528740475009479946108914\ 531695974662467316502299096287428156173285763007 2 Pi Since this is negative, there is no Apery-style irrationality proof of, ---, 24 but still a very fast way to compute it to many digits ----------------------- This took, 5.391, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 3 k ) binomial(n, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ 3 k b(n) = ) binomial(n, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 3 2 3 (9 n - 24 n + 17 n - 4) b(n - 1) b(n) = ------------------------------------ 2 n (3 n - 5) 2 2 3 (3 n - 4) (9 n - 21 n + 11) b(n - 2) 27 (3 n - 2) (n - 2) b(n - 3) + --------------------------------------- + ------------------------------ 2 2 n (3 n - 5) n (3 n - 5) and in Maple notation b(n) = 3*(9*n^3-24*n^2+17*n-4)/n^2/(3*n-5)*b(n-1)+3*(3*n-4)*(9*n^2-21*n+11)/n^2 /(3*n-5)*b(n-2)+27*(3*n-2)*(n-2)^2/n^2/(3*n-5)*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 3, b(2) = 21 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 3 2 3 (9 n - 24 n + 17 n - 4) a(n - 1) a(n) = ------------------------------------ 2 n (3 n - 5) 2 2 3 (3 n - 4) (9 n - 21 n + 11) a(n - 2) 27 (3 n - 2) (n - 2) a(n - 3) + --------------------------------------- + ------------------------------ 2 2 n (3 n - 5) n (3 n - 5) but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.23104906018664843647241070715272552269183337812008508470689333649779787398\ 98982385352877756654728958 This constant is identified as, 1/3 ln(2) The implied delta is, -0.4098897689679997615255986678361674616682969553806003\ 348985641023540612923720261266630744982636388650 Since this is negative, there is no Apery-style irrationality proof of, 1/3 ln(2), but still a very fast way to compute it to many digits ----------------------- This took, 0.853, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 3 k ) binomial(n, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ 3 k b(n) = ) binomial(n, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 3 2 4 (9 n - 24 n + 17 n - 4) b(n - 1) b(n) = ------------------------------------ 2 n (3 n - 5) 2 2 (3 n - 4) (33 n - 77 n + 42) b(n - 2) 64 (3 n - 2) (n - 2) b(n - 3) + -------------------------------------- + ------------------------------ 2 2 n (3 n - 5) n (3 n - 5) and in Maple notation b(n) = 4*(9*n^3-24*n^2+17*n-4)/n^2/(3*n-5)*b(n-1)+(3*n-4)*(33*n^2-77*n+42)/n^2/ (3*n-5)*b(n-2)+64*(3*n-2)*(n-2)^2/n^2/(3*n-5)*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 4, b(2) = 34 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 3 2 4 (9 n - 24 n + 17 n - 4) a(n - 1) a(n) = ------------------------------------ 2 n (3 n - 5) 2 2 (3 n - 4) (33 n - 77 n + 42) a(n - 2) 64 (3 n - 2) (n - 2) a(n - 3) + -------------------------------------- + ------------------------------ 2 2 n (3 n - 5) n (3 n - 5) but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.18310204811135161523254087282042095077458175963712490862244905560624904886\ 97681611456026258022886815 This constant is identified as, 1/6 ln(3) The implied delta is, -0.4697795379202535650594372488544484427821791878504795\ 230507156743760012361444529201032742854928399279 Since this is negative, there is no Apery-style irrationality proof of, 1/6 ln(3), but still a very fast way to compute it to many digits ----------------------- This took, 0.953, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 ) binomial(n, k) binomial(n + k, k) / ----- k By Shalosh B. Ekhad Let ----- \ 2 b(n) = ) binomial(n, k) binomial(n + k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 2 2 (11 n - 11 n + 3) b(n - 1) (n - 1) b(n - 2) b(n) = --------------------------- + ----------------- 2 2 n n and in Maple notation b(n) = (11*n^2-11*n+3)/n^2*b(n-1)+(n-1)^2/n^2*b(n-2) Of course, the initial conditions are b(0) = 1, b(1) = 3 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 2 2 (11 n - 11 n + 3) a(n - 1) (n - 1) a(n - 2) a(n) = --------------------------- + ----------------- 2 2 n n but with the following simpler initial conditions a(0) = 0, a(1) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.32898681336964528729448303332920503784378998024135968754711164587400149408\ 06401747667257801239517411 2 Pi This constant is identified as, --- 30 The implied delta is, 0.10027863510256443223790034039381962411836408279873220\ 6361537750094227943137990180700698208898802552 Since this is positive, this suggests an Apery-style irrationality proof of, 2 Pi --- 30 ----------------------- This took, 3.889, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(n + k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(n + k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 3 2 (703 n - 1881 n + 1383 n - 330) b(n - 1) b(n) = ------------------------------------------ 2 n (37 n - 62) 3 2 (999 n - 3672 n + 4267 n - 1510) b(n - 2) - ------------------------------------------- 2 n (37 n - 62) 2 (37 n - 25) (n - 2) b(n - 3) - ----------------------------- 2 n (37 n - 62) and in Maple notation b(n) = (703*n^3-1881*n^2+1383*n-330)/n^2/(37*n-62)*b(n-1)-(999*n^3-3672*n^2+ 4267*n-1510)/n^2/(37*n-62)*b(n-2)-(37*n-25)*(n-2)^2/n^2/(37*n-62)*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 5, b(2) = 49 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 3 2 (703 n - 1881 n + 1383 n - 330) a(n - 1) a(n) = ------------------------------------------ 2 n (37 n - 62) 3 2 (999 n - 3672 n + 4267 n - 1510) a(n - 2) - ------------------------------------------- 2 n (37 n - 62) 2 (37 n - 25) (n - 2) a(n - 3) - ----------------------------- 2 n (37 n - 62) but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.23104906018664843647241070715272552269183337812008508470689333649779787398\ 98982385352877756654728958 This constant is identified as, 1/3 ln(2) The implied delta is, -0.3761432057026443817070711846776682924819129238984229\ 211497801799338352617158681689833762490800357400 Since this is negative, there is no Apery-style irrationality proof of, 1/3 ln(2), but still a very fast way to compute it to many digits ----------------------- This took, 0.811, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(n + k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(n + k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 3 2 (621 n - 1647 n + 1211 n - 290) b(n - 1) b(n) = ------------------------------------------ 2 n (23 n - 38) 3 2 (2001 n - 7308 n + 8413 n - 2930) b(n - 2) - -------------------------------------------- 2 n (23 n - 38) 2 2 (23 n - 15) (n - 2) b(n - 3) - ------------------------------- 2 n (23 n - 38) and in Maple notation b(n) = (621*n^3-1647*n^2+1211*n-290)/n^2/(23*n-38)*b(n-1)-(2001*n^3-7308*n^2+ 8413*n-2930)/n^2/(23*n-38)*b(n-2)-2*(23*n-15)*(n-2)^2/n^2/(23*n-38)*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 7, b(2) = 91 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 3 2 (621 n - 1647 n + 1211 n - 290) a(n - 1) a(n) = ------------------------------------------ 2 n (23 n - 38) 3 2 (2001 n - 7308 n + 8413 n - 2930) a(n - 2) - -------------------------------------------- 2 n (23 n - 38) 2 2 (23 n - 15) (n - 2) a(n - 3) - ------------------------------- 2 n (23 n - 38) but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.18310204811135161523254087282042095077458175963712490862244905560624904886\ 97681611456026258022886815 This constant is identified as, 1/6 ln(3) The implied delta is, -0.5587274206085831095803288947824203524366916864370397\ 300340047920805004646843897963490650296125843514 Since this is negative, there is no Apery-style irrationality proof of, 1/6 ln(3), but still a very fast way to compute it to many digits ----------------------- This took, 0.919, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 ) binomial(n, k) binomial(n + 2 k, k) / ----- k By Shalosh B. Ekhad Let ----- \ 2 b(n) = ) binomial(n, k) binomial(n + 2 k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 9 8 7 6 b(n) = 1/4 (54053883 n - 618518583 n + 3010474698 n - 8144063592 n 5 4 3 2 + 13437822349 n - 13979663551 n + 9161269990 n - 3656805666 n / 2 2 + 811372464 n - 76966344) b(n - 1) / (n %1 (2 n - 1) ) - 1/4 ( / 9 8 7 6 89261393 n - 1110646486 n + 5961590769 n - 18068217810 n 5 4 3 2 + 33959420883 n - 40874842244 n + 31342153523 n - 14672618280 n / 2 2 + 3782942636 n - 409156272) b(n - 2) / (n %1 (2 n - 1) ) + 1/4 ( / 7 6 5 4 3 5384678 n - 50845522 n + 192846201 n - 378502842 n + 413658206 n 2 2 / 2 - 250904342 n + 78074567 n - 9713970) (n - 2) b(n - 3) / (n %1 / 2 (2 n - 1) ) - 9/4 5 4 3 2 (207103 n - 920082 n + 1554201 n - 1235568 n + 457990 n - 64022) 2 2 / 2 2 (n - 2) (n - 3) b(n - 4) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 207103 n - 1955597 n + 7305559 n - 13489693 n + 12307572 n - 4438966 and in Maple notation b(n) = 1/4*(54053883*n^9-618518583*n^8+3010474698*n^7-8144063592*n^6+ 13437822349*n^5-13979663551*n^4+9161269990*n^3-3656805666*n^2+811372464*n-\ 76966344)/n^2/(207103*n^5-1955597*n^4+7305559*n^3-13489693*n^2+12307572*n-\ 4438966)/(2*n-1)^2*b(n-1)-1/4*(89261393*n^9-1110646486*n^8+5961590769*n^7-\ 18068217810*n^6+33959420883*n^5-40874842244*n^4+31342153523*n^3-14672618280*n^2 +3782942636*n-409156272)/n^2/(207103*n^5-1955597*n^4+7305559*n^3-13489693*n^2+ 12307572*n-4438966)/(2*n-1)^2*b(n-2)+1/4*(5384678*n^7-50845522*n^6+192846201*n^ 5-378502842*n^4+413658206*n^3-250904342*n^2+78074567*n-9713970)*(n-2)^2/n^2/( 207103*n^5-1955597*n^4+7305559*n^3-13489693*n^2+12307572*n-4438966)/(2*n-1)^2*b (n-3)-9/4*(207103*n^5-920082*n^4+1554201*n^3-1235568*n^2+457990*n-64022)*(n-2)^ 2*(n-3)^2/n^2/(207103*n^5-1955597*n^4+7305559*n^3-13489693*n^2+12307572*n-\ 4438966)/(2*n-1)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 4, b(2) = 32, b(3) = 319 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 9 8 7 6 a(n) = 1/4 (54053883 n - 618518583 n + 3010474698 n - 8144063592 n 5 4 3 2 + 13437822349 n - 13979663551 n + 9161269990 n - 3656805666 n / 2 2 + 811372464 n - 76966344) a(n - 1) / (n %1 (2 n - 1) ) - 1/4 ( / 9 8 7 6 89261393 n - 1110646486 n + 5961590769 n - 18068217810 n 5 4 3 2 + 33959420883 n - 40874842244 n + 31342153523 n - 14672618280 n / 2 2 + 3782942636 n - 409156272) a(n - 2) / (n %1 (2 n - 1) ) + 1/4 ( / 7 6 5 4 3 5384678 n - 50845522 n + 192846201 n - 378502842 n + 413658206 n 2 2 / 2 - 250904342 n + 78074567 n - 9713970) (n - 2) a(n - 3) / (n %1 / 2 (2 n - 1) ) - 9/4 5 4 3 2 (207103 n - 920082 n + 1554201 n - 1235568 n + 457990 n - 64022) 2 2 / 2 2 (n - 2) (n - 3) a(n - 4) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 207103 n - 1955597 n + 7305559 n - 13489693 n + 12307572 n - 4438966 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.15451926014658719908930444060693572105619882479207788707139271204700983489\ -1805 38695700882306867480684197 10 This constant is identified as, 0 The implied delta is, -0.5531517047101389183755373306834948486550937041829726\ 077398117487345249381602345304197253572055719780 Since this is negative, there is no Apery-style irrationality proof of, 0, but still a very fast way to compute it to many digits ----------------------- This took, 1.886, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(n + 2 k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(n + 2 k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 9 8 7 b(n) = 1/8 (11565805271 n - 131961700340 n + 640549005508 n 6 5 4 - 1728914653136 n + 2848443856247 n - 2961948339624 n 3 2 + 1942512252110 n - 776830448596 n + 172828941120 n - 16443705600) / 2 2 9 8 b(n - 1) / (n %1 (2 n - 1) ) - 1/8 (54989153917 n - 682395727097 n / 7 6 5 + 3651960599461 n - 11030361916253 n + 20648908563060 n 4 3 2 - 24737635408058 n + 18866211068334 n - 8779693495188 n / 2 2 + 2250010065760 n - 241882636800) b(n - 2) / (n %1 (2 n - 1) ) + 1/8 ( / 7 6 5 4 1468481047 n - 13817881286 n + 52081746555 n - 101329547743 n 3 2 2 + 109599843654 n - 65760223043 n + 20251335604 n - 2491924260) (n - 2) / 2 2 5 4 b(n - 3) / (n %1 (2 n - 1) ) - 49/8 (12136207 n - 53516331 n / 3 2 2 2 + 89472959 n - 70214027 n + 25707076 n - 3553340) (n - 2) (n - 3) / 2 2 b(n - 4) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 12136207 n - 114197366 n + 424900353 n - 781092960 n + 709300366 n - 254599940 and in Maple notation b(n) = 1/8*(11565805271*n^9-131961700340*n^8+640549005508*n^7-1728914653136*n^6 +2848443856247*n^5-2961948339624*n^4+1942512252110*n^3-776830448596*n^2+ 172828941120*n-16443705600)/n^2/(12136207*n^5-114197366*n^4+424900353*n^3-\ 781092960*n^2+709300366*n-254599940)/(2*n-1)^2*b(n-1)-1/8*(54989153917*n^9-\ 682395727097*n^8+3651960599461*n^7-11030361916253*n^6+20648908563060*n^5-\ 24737635408058*n^4+18866211068334*n^3-8779693495188*n^2+2250010065760*n-\ 241882636800)/n^2/(12136207*n^5-114197366*n^4+424900353*n^3-781092960*n^2+ 709300366*n-254599940)/(2*n-1)^2*b(n-2)+1/8*(1468481047*n^7-13817881286*n^6+ 52081746555*n^5-101329547743*n^4+109599843654*n^3-65760223043*n^2+20251335604*n -2491924260)*(n-2)^2/n^2/(12136207*n^5-114197366*n^4+424900353*n^3-781092960*n^ 2+709300366*n-254599940)/(2*n-1)^2*b(n-3)-49/8*(12136207*n^5-53516331*n^4+ 89472959*n^3-70214027*n^2+25707076*n-3553340)*(n-2)^2*(n-3)^2/n^2/(12136207*n^5 -114197366*n^4+424900353*n^3-781092960*n^2+709300366*n-254599940)/(2*n-1)^2*b(n -4) Of course, the initial conditions are b(0) = 1, b(1) = 7, b(2) = 93, b(3) = 1519 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 9 8 7 a(n) = 1/8 (11565805271 n - 131961700340 n + 640549005508 n 6 5 4 - 1728914653136 n + 2848443856247 n - 2961948339624 n 3 2 + 1942512252110 n - 776830448596 n + 172828941120 n - 16443705600) / 2 2 9 8 a(n - 1) / (n %1 (2 n - 1) ) - 1/8 (54989153917 n - 682395727097 n / 7 6 5 + 3651960599461 n - 11030361916253 n + 20648908563060 n 4 3 2 - 24737635408058 n + 18866211068334 n - 8779693495188 n / 2 2 + 2250010065760 n - 241882636800) a(n - 2) / (n %1 (2 n - 1) ) + 1/8 ( / 7 6 5 4 1468481047 n - 13817881286 n + 52081746555 n - 101329547743 n 3 2 2 + 109599843654 n - 65760223043 n + 20251335604 n - 2491924260) (n - 2) / 2 2 5 4 a(n - 3) / (n %1 (2 n - 1) ) - 49/8 (12136207 n - 53516331 n / 3 2 2 2 + 89472959 n - 70214027 n + 25707076 n - 3553340) (n - 2) (n - 3) / 2 2 a(n - 4) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 12136207 n - 114197366 n + 424900353 n - 781092960 n + 709300366 n - 254599940 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.34657359027997265470861606072908828403775006718012762706034000474669681098\ 48473578029316634982093438 This constant is identified as, 1/2 ln(2) The implied delta is, -0.7292218309842496641892218664098147725415438432362439\ 332533906650971960984773755589241626043236915564 Since this is negative, there is no Apery-style irrationality proof of, 1/2 ln(2), but still a very fast way to compute it to many digits ----------------------- This took, 2.269, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(n + 2 k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(n + 2 k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 9 8 7 b(n) = 1/12 (76274071861 n - 869390051439 n + 4216366304020 n 6 5 4 - 11372891532002 n + 18730580535525 n - 19477761112693 n 3 2 + 12780066264538 n - 5115343732074 n + 1139367409056 n - 108538191432) / 2 2 9 8 b(n - 1) / (n %1 (2 n - 1) ) - 1/12 (612432689661 n - 7593086463300 n / 7 6 5 + 40594306718901 n - 122469661443116 n + 228962041294173 n 4 3 2 - 273884864454952 n + 208522408116713 n - 96860128482228 n / 2 2 + 24776978186356 n - 2658712114080) b(n - 2) / (n %1 (2 n - 1) ) + 1/12 / 7 6 5 4 (10576279584 n - 99398396448 n + 373878207417 n - 725329159750 n 3 2 + 781846829564 n - 467423857316 n + 143461797727 n - 17591735898) 2 / 2 2 121 5 4 (n - 2) b(n - 3) / (n %1 (2 n - 1) ) - --- (36723193 n - 161517356 n / 12 3 2 2 2 + 269168185 n - 210452554 n + 76796776 n - 10583622) (n - 2) (n - 3) / 2 2 b(n - 4) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 36723193 n - 345133321 n + 1282469539 n - 2354293175 n + 2134891828 n - 765241686 and in Maple notation b(n) = 1/12*(76274071861*n^9-869390051439*n^8+4216366304020*n^7-11372891532002* n^6+18730580535525*n^5-19477761112693*n^4+12780066264538*n^3-5115343732074*n^2+ 1139367409056*n-108538191432)/n^2/(36723193*n^5-345133321*n^4+1282469539*n^3-\ 2354293175*n^2+2134891828*n-765241686)/(2*n-1)^2*b(n-1)-1/12*(612432689661*n^9-\ 7593086463300*n^8+40594306718901*n^7-122469661443116*n^6+228962041294173*n^5-\ 273884864454952*n^4+208522408116713*n^3-96860128482228*n^2+24776978186356*n-\ 2658712114080)/n^2/(36723193*n^5-345133321*n^4+1282469539*n^3-2354293175*n^2+ 2134891828*n-765241686)/(2*n-1)^2*b(n-2)+1/12*(10576279584*n^7-99398396448*n^6+ 373878207417*n^5-725329159750*n^4+781846829564*n^3-467423857316*n^2+ 143461797727*n-17591735898)*(n-2)^2/n^2/(36723193*n^5-345133321*n^4+1282469539* n^3-2354293175*n^2+2134891828*n-765241686)/(2*n-1)^2*b(n-3)-121/12*(36723193*n^ 5-161517356*n^4+269168185*n^3-210452554*n^2+76796776*n-10583622)*(n-2)^2*(n-3)^ 2/n^2/(36723193*n^5-345133321*n^4+1282469539*n^3-2354293175*n^2+2134891828*n-\ 765241686)/(2*n-1)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 10, b(2) = 184, b(3) = 4105 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 9 8 7 a(n) = 1/12 (76274071861 n - 869390051439 n + 4216366304020 n 6 5 4 - 11372891532002 n + 18730580535525 n - 19477761112693 n 3 2 + 12780066264538 n - 5115343732074 n + 1139367409056 n - 108538191432) / 2 2 9 8 a(n - 1) / (n %1 (2 n - 1) ) - 1/12 (612432689661 n - 7593086463300 n / 7 6 5 + 40594306718901 n - 122469661443116 n + 228962041294173 n 4 3 2 - 273884864454952 n + 208522408116713 n - 96860128482228 n / 2 2 + 24776978186356 n - 2658712114080) a(n - 2) / (n %1 (2 n - 1) ) + 1/12 / 7 6 5 4 (10576279584 n - 99398396448 n + 373878207417 n - 725329159750 n 3 2 + 781846829564 n - 467423857316 n + 143461797727 n - 17591735898) 2 / 2 2 121 5 4 (n - 2) a(n - 3) / (n %1 (2 n - 1) ) - --- (36723193 n - 161517356 n / 12 3 2 2 2 + 269168185 n - 210452554 n + 76796776 n - 10583622) (n - 2) (n - 3) / 2 2 a(n - 4) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 36723193 n - 345133321 n + 1282469539 n - 2354293175 n + 2134891828 n - 765241686 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.24413606414846882031005449709389460103277567951616654482993207414166539849\ 30242148608035010697182420 This constant is identified as, 2/9 ln(3) The implied delta is, -0.7949219923570801537590992425684979973495072155357806\ 487317249901233331176410377332074613298722620502 Since this is negative, there is no Apery-style irrationality proof of, 2/9 ln(3), but still a very fast way to compute it to many digits ----------------------- This took, 2.415, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 ) binomial(n, k) binomial(n + k, k) / ----- k By Shalosh B. Ekhad Let ----- \ 2 b(n) = ) binomial(n, k) binomial(n + k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 2 2 (11 n - 11 n + 3) b(n - 1) (n - 1) b(n - 2) b(n) = --------------------------- + ----------------- 2 2 n n and in Maple notation b(n) = (11*n^2-11*n+3)/n^2*b(n-1)+(n-1)^2/n^2*b(n-2) Of course, the initial conditions are b(0) = 1, b(1) = 3 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 2 2 (11 n - 11 n + 3) a(n - 1) (n - 1) a(n - 2) a(n) = --------------------------- + ----------------- 2 2 n n but with the following simpler initial conditions a(0) = 0, a(1) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.32898681336964528729448303332920503784378998024135968754711164587400149408\ 06401747667257801239517411 2 Pi This constant is identified as, --- 30 The implied delta is, 0.10027863510256443223790034039381962411836408279873220\ 6361537750094227943137990180700698208898802552 Since this is positive, this suggests an Apery-style irrationality proof of, 2 Pi --- 30 ----------------------- This took, 3.227, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(n + k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(n + k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 3 2 (703 n - 1881 n + 1383 n - 330) b(n - 1) b(n) = ------------------------------------------ 2 n (37 n - 62) 3 2 (999 n - 3672 n + 4267 n - 1510) b(n - 2) - ------------------------------------------- 2 n (37 n - 62) 2 (37 n - 25) (n - 2) b(n - 3) - ----------------------------- 2 n (37 n - 62) and in Maple notation b(n) = (703*n^3-1881*n^2+1383*n-330)/n^2/(37*n-62)*b(n-1)-(999*n^3-3672*n^2+ 4267*n-1510)/n^2/(37*n-62)*b(n-2)-(37*n-25)*(n-2)^2/n^2/(37*n-62)*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 5, b(2) = 49 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 3 2 (703 n - 1881 n + 1383 n - 330) a(n - 1) a(n) = ------------------------------------------ 2 n (37 n - 62) 3 2 (999 n - 3672 n + 4267 n - 1510) a(n - 2) - ------------------------------------------- 2 n (37 n - 62) 2 (37 n - 25) (n - 2) a(n - 3) - ----------------------------- 2 n (37 n - 62) but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.23104906018664843647241070715272552269183337812008508470689333649779787398\ 98982385352877756654728958 This constant is identified as, 1/3 ln(2) The implied delta is, -0.3761432057026443817070711846776682924819129238984229\ 211497801799338352617158681689833762490800357400 Since this is negative, there is no Apery-style irrationality proof of, 1/3 ln(2), but still a very fast way to compute it to many digits ----------------------- This took, 0.382, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(n + k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(n + k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 3 2 (621 n - 1647 n + 1211 n - 290) b(n - 1) b(n) = ------------------------------------------ 2 n (23 n - 38) 3 2 (2001 n - 7308 n + 8413 n - 2930) b(n - 2) - -------------------------------------------- 2 n (23 n - 38) 2 2 (23 n - 15) (n - 2) b(n - 3) - ------------------------------- 2 n (23 n - 38) and in Maple notation b(n) = (621*n^3-1647*n^2+1211*n-290)/n^2/(23*n-38)*b(n-1)-(2001*n^3-7308*n^2+ 8413*n-2930)/n^2/(23*n-38)*b(n-2)-2*(23*n-15)*(n-2)^2/n^2/(23*n-38)*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 7, b(2) = 91 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 3 2 (621 n - 1647 n + 1211 n - 290) a(n - 1) a(n) = ------------------------------------------ 2 n (23 n - 38) 3 2 (2001 n - 7308 n + 8413 n - 2930) a(n - 2) - -------------------------------------------- 2 n (23 n - 38) 2 2 (23 n - 15) (n - 2) a(n - 3) - ------------------------------- 2 n (23 n - 38) but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.18310204811135161523254087282042095077458175963712490862244905560624904886\ 97681611456026258022886815 This constant is identified as, 1/6 ln(3) The implied delta is, -0.5587274206085831095803288947824203524366916864370397\ 300340047920805004646843897963490650296125843514 Since this is negative, there is no Apery-style irrationality proof of, 1/6 ln(3), but still a very fast way to compute it to many digits ----------------------- This took, 0.631, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 ) binomial(n, k) binomial(n + k, k) / ----- k By Shalosh B. Ekhad Let ----- \ 2 b(n) = ) binomial(n, k) binomial(n + k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 3 2 (2301 n - 5538 n + 3797 n - 800) b(n - 1) b(n) = 1/2 ------------------------------------------- 2 n (59 n - 83) 3 2 (59 n - 201 n + 213 n - 64) b(n - 2) + 5/2 -------------------------------------- 2 n (59 n - 83) 2 (59 n - 24) (n - 2) b(n - 3) + 1/2 ----------------------------- 2 n (59 n - 83) and in Maple notation b(n) = 1/2*(2301*n^3-5538*n^2+3797*n-800)/n^2/(59*n-83)*b(n-1)+5/2*(59*n^3-201* n^2+213*n-64)/n^2/(59*n-83)*b(n-2)+1/2*(59*n-24)*(n-2)^2/n^2/(59*n-83)*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 5, b(2) = 55 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 3 2 (2301 n - 5538 n + 3797 n - 800) a(n - 1) a(n) = 1/2 ------------------------------------------- 2 n (59 n - 83) 3 2 (59 n - 201 n + 213 n - 64) a(n - 2) + 5/2 -------------------------------------- 2 n (59 n - 83) 2 (59 n - 24) (n - 2) a(n - 3) + 1/2 ----------------------------- 2 n (59 n - 83) but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.19804205158855580269063774898805044802157146696007292974876571699811246341\ 99127758873895219989767679 This constant is identified as, 2/7 ln(2) The implied delta is, 0.03573063624714966655055172016617059903249715199244778\ 0934019797491619619000229222863713450858530549 Since this is positive, this suggests an Apery-style irrationality proof of, 2/7 ln(2) ----------------------- This took, 1.261, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(n + k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(n + k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 3 2 (9737 n - 23326 n + 16001 n - 3384) b(n - 1) b(n) = 1/3 ---------------------------------------------- 2 n (91 n - 127) 3 2 (1183 n - 4017 n + 4175 n - 1176) b(n - 2) + 1/3 -------------------------------------------- 2 n (91 n - 127) 2 (91 n - 36) (n - 2) b(n - 3) + 1/3 ----------------------------- 2 n (91 n - 127) and in Maple notation b(n) = 1/3*(9737*n^3-23326*n^2+16001*n-3384)/n^2/(91*n-127)*b(n-1)+1/3*(1183*n^ 3-4017*n^2+4175*n-1176)/n^2/(91*n-127)*b(n-2)+1/3*(91*n-36)*(n-2)^2/n^2/(91*n-\ 127)*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 9, b(2) = 181 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 3 2 (9737 n - 23326 n + 16001 n - 3384) a(n - 1) a(n) = 1/3 ---------------------------------------------- 2 n (91 n - 127) 3 2 (1183 n - 4017 n + 4175 n - 1176) a(n - 2) + 1/3 -------------------------------------------- 2 n (91 n - 127) 2 (91 n - 36) (n - 2) a(n - 3) + 1/3 ----------------------------- 2 n (91 n - 127) but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.11058139312040846781218539512664067361054284276249841753109481567566381943\ 15220685275688439501763822 This constant is identified as, 3/11 ln(3) - 3/11 ln(2) The implied delta is, 0.04615348752875711003555417864556679856262699940926071\ 7644317018386028234628322472030887530330604894 Since this is positive, this suggests an Apery-style irrationality proof of, 3/11 ln(3) - 3/11 ln(2) ----------------------- This took, 1.340, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(n + k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(n + k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 3 2 (8487 n - 20286 n + 13911 n - 2944) b(n - 1) b(n) = 1/4 ---------------------------------------------- 2 n (41 n - 57) 3 2 (861 n - 2919 n + 3015 n - 832) b(n - 2) + 1/4 ------------------------------------------ 2 n (41 n - 57) 2 (n - 2) (41 n - 16) b(n - 3) + 1/4 ----------------------------- 2 n (41 n - 57) and in Maple notation b(n) = 1/4*(8487*n^3-20286*n^2+13911*n-2944)/n^2/(41*n-57)*b(n-1)+1/4*(861*n^3-\ 2919*n^2+3015*n-832)/n^2/(41*n-57)*b(n-2)+1/4*(n-2)^2*(41*n-16)/n^2/(41*n-57)*b (n-3) Of course, the initial conditions are b(0) = 1, b(1) = 13, b(2) = 379 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 3 2 (8487 n - 20286 n + 13911 n - 2944) a(n - 1) a(n) = 1/4 ---------------------------------------------- 2 n (41 n - 57) 3 2 (861 n - 2919 n + 3015 n - 832) a(n - 2) + 1/4 ------------------------------------------ 2 n (41 n - 57) 2 (n - 2) (41 n - 16) a(n - 3) + 1/4 ----------------------------- 2 n (41 n - 57) but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.07671521932047491398379173493168731506760258957273628173511084942647812019\ 220812382349623978109474301 This constant is identified as, 8/15 ln(2) - 4/15 ln(3) The implied delta is, 0.04841419362294527821702422246395289617737416741001549\ 5206688170813711949147492936955810750027730494 Since this is positive, this suggests an Apery-style irrationality proof of, 8/15 ln(2) - 4/15 ln(3) ----------------------- This took, 1.528, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ ) binomial(n, k) binomial(n + k, k) binomial(n + 2 k, k) / ----- k By Shalosh B. Ekhad Let ----- \ b(n) = ) binomial(n, k) binomial(n + k, k) binomial(n + 2 k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 2 (3 n - 2) (30 n - 50 n + 13) b(n - 1) b(n) = -------------------------------------- 2 n (3 n - 4) 3 2 2 (9 n - 30 n + 29 n - 6) b(n - 2) (3 n - 1) (n - 2) b(n - 3) - ---------------------------------- + --------------------------- 2 2 n (3 n - 4) n (3 n - 4) and in Maple notation b(n) = (3*n-2)*(30*n^2-50*n+13)/n^2/(3*n-4)*b(n-1)-(9*n^3-30*n^2+29*n-6)/n^2/(3 *n-4)*b(n-2)+(3*n-1)*(n-2)^2/n^2/(3*n-4)*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 7, b(2) = 115 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 2 (3 n - 2) (30 n - 50 n + 13) a(n - 1) a(n) = -------------------------------------- 2 n (3 n - 4) 3 2 2 (9 n - 30 n + 29 n - 6) a(n - 2) (3 n - 1) (n - 2) a(n - 3) - ---------------------------------- + --------------------------- 2 2 n (3 n - 4) n (3 n - 4) but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.14346964698800134629647774431686847514152356084676321825233536799752040515\ 37106537871634215657399851 The implied delta is, 0.16444191198415579425338050557553390927872515945932884\ 7178334036407146818419916808577331206271324100 Since this is positive, this suggests an Apery-style irrationality proof of, 0.1434696469880013462964777443168684751415235608467632182523353679975204\ 051537106537871634215657399851 ----------------------- This took, 5.585, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + k, k) binomial(n + 2 k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + k, k) binomial(n + 2 k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 2 (3 n - 2) (57 n - 95 n + 25) b(n - 1) b(n) = -------------------------------------- 2 n (3 n - 4) 3 2 2 (9 n - 30 n + 29 n - 6) b(n - 2) (3 n - 1) (n - 2) b(n - 3) - ---------------------------------- + --------------------------- 2 2 n (3 n - 4) n (3 n - 4) and in Maple notation b(n) = (3*n-2)*(57*n^2-95*n+25)/n^2/(3*n-4)*b(n-1)-(9*n^3-30*n^2+29*n-6)/n^2/(3 *n-4)*b(n-2)+(3*n-1)*(n-2)^2/n^2/(3*n-4)*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 13, b(2) = 409 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 2 (3 n - 2) (57 n - 95 n + 25) a(n - 1) a(n) = -------------------------------------- 2 n (3 n - 4) 3 2 2 (9 n - 30 n + 29 n - 6) a(n - 2) (3 n - 1) (n - 2) a(n - 3) - ---------------------------------- + --------------------------- 2 2 n (3 n - 4) n (3 n - 4) but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.07701635339554947882413690238424184089727779270669502823563111216593262466\ 329941284509592522182429862 This constant is identified as, 1/9 ln(2) The implied delta is, 0.20778374377590423771128437544757387878100294839652882\ 5775879219790823183644112529390443490682262721 Since this is positive, this suggests an Apery-style irrationality proof of, 1/9 ln(2) ----------------------- This took, 1.646, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + k, k) binomial(n + 2 k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + k, k) binomial(n + 2 k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 2 (3 n - 2) (84 n - 140 n + 37) b(n - 1) b(n) = --------------------------------------- 2 n (3 n - 4) 3 2 2 (9 n - 30 n + 29 n - 6) b(n - 2) (3 n - 1) (n - 2) b(n - 3) - ---------------------------------- + --------------------------- 2 2 n (3 n - 4) n (3 n - 4) and in Maple notation b(n) = (3*n-2)*(84*n^2-140*n+37)/n^2/(3*n-4)*b(n-1)-(9*n^3-30*n^2+29*n-6)/n^2/( 3*n-4)*b(n-2)+(3*n-1)*(n-2)^2/n^2/(3*n-4)*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 19, b(2) = 883 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 2 (3 n - 2) (84 n - 140 n + 37) a(n - 1) a(n) = --------------------------------------- 2 n (3 n - 4) 3 2 2 (9 n - 30 n + 29 n - 6) a(n - 2) (3 n - 1) (n - 2) a(n - 3) - ---------------------------------- + --------------------------- 2 2 n (3 n - 4) n (3 n - 4) but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.05266121197729926435149028103707346284008790617657377777528917812110456158\ 083182257495374843300288516 The implied delta is, 0.22920459514683997933787709019996781555419456269156642\ 0367269597988852353845153933422933946923027391 Since this is positive, this suggests an Apery-style irrationality proof of, 0.0526612119772992643514902810370734628400879061765737777752891781211045\ 6158083182257495374843300288516 ----------------------- This took, 4.168, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 ) binomial(n, k) binomial(n + 2 k, k) / ----- k By Shalosh B. Ekhad Let ----- \ 2 b(n) = ) binomial(n, k) binomial(n + 2 k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 9 8 7 6 b(n) = 1/4 (54053883 n - 618518583 n + 3010474698 n - 8144063592 n 5 4 3 2 + 13437822349 n - 13979663551 n + 9161269990 n - 3656805666 n / 2 2 + 811372464 n - 76966344) b(n - 1) / (n %1 (2 n - 1) ) - 1/4 ( / 9 8 7 6 89261393 n - 1110646486 n + 5961590769 n - 18068217810 n 5 4 3 2 + 33959420883 n - 40874842244 n + 31342153523 n - 14672618280 n / 2 2 + 3782942636 n - 409156272) b(n - 2) / (n %1 (2 n - 1) ) + 1/4 ( / 7 6 5 4 3 5384678 n - 50845522 n + 192846201 n - 378502842 n + 413658206 n 2 2 / 2 - 250904342 n + 78074567 n - 9713970) (n - 2) b(n - 3) / (n %1 / 2 (2 n - 1) ) - 9/4 5 4 3 2 (207103 n - 920082 n + 1554201 n - 1235568 n + 457990 n - 64022) 2 2 / 2 2 (n - 2) (n - 3) b(n - 4) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 207103 n - 1955597 n + 7305559 n - 13489693 n + 12307572 n - 4438966 and in Maple notation b(n) = 1/4*(54053883*n^9-618518583*n^8+3010474698*n^7-8144063592*n^6+ 13437822349*n^5-13979663551*n^4+9161269990*n^3-3656805666*n^2+811372464*n-\ 76966344)/n^2/(207103*n^5-1955597*n^4+7305559*n^3-13489693*n^2+12307572*n-\ 4438966)/(2*n-1)^2*b(n-1)-1/4*(89261393*n^9-1110646486*n^8+5961590769*n^7-\ 18068217810*n^6+33959420883*n^5-40874842244*n^4+31342153523*n^3-14672618280*n^2 +3782942636*n-409156272)/n^2/(207103*n^5-1955597*n^4+7305559*n^3-13489693*n^2+ 12307572*n-4438966)/(2*n-1)^2*b(n-2)+1/4*(5384678*n^7-50845522*n^6+192846201*n^ 5-378502842*n^4+413658206*n^3-250904342*n^2+78074567*n-9713970)*(n-2)^2/n^2/( 207103*n^5-1955597*n^4+7305559*n^3-13489693*n^2+12307572*n-4438966)/(2*n-1)^2*b (n-3)-9/4*(207103*n^5-920082*n^4+1554201*n^3-1235568*n^2+457990*n-64022)*(n-2)^ 2*(n-3)^2/n^2/(207103*n^5-1955597*n^4+7305559*n^3-13489693*n^2+12307572*n-\ 4438966)/(2*n-1)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 4, b(2) = 32, b(3) = 319 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 9 8 7 6 a(n) = 1/4 (54053883 n - 618518583 n + 3010474698 n - 8144063592 n 5 4 3 2 + 13437822349 n - 13979663551 n + 9161269990 n - 3656805666 n / 2 2 + 811372464 n - 76966344) a(n - 1) / (n %1 (2 n - 1) ) - 1/4 ( / 9 8 7 6 89261393 n - 1110646486 n + 5961590769 n - 18068217810 n 5 4 3 2 + 33959420883 n - 40874842244 n + 31342153523 n - 14672618280 n / 2 2 + 3782942636 n - 409156272) a(n - 2) / (n %1 (2 n - 1) ) + 1/4 ( / 7 6 5 4 3 5384678 n - 50845522 n + 192846201 n - 378502842 n + 413658206 n 2 2 / 2 - 250904342 n + 78074567 n - 9713970) (n - 2) a(n - 3) / (n %1 / 2 (2 n - 1) ) - 9/4 5 4 3 2 (207103 n - 920082 n + 1554201 n - 1235568 n + 457990 n - 64022) 2 2 / 2 2 (n - 2) (n - 3) a(n - 4) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 207103 n - 1955597 n + 7305559 n - 13489693 n + 12307572 n - 4438966 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.15451926014658719908930444060693572105619882479207788707139271204700983489\ -1805 38695700882306867480684197 10 This constant is identified as, 0 The implied delta is, -0.5531517047101389183755373306834948486550937041829726\ 077398117487345249381602345304197253572055719780 Since this is negative, there is no Apery-style irrationality proof of, 0, but still a very fast way to compute it to many digits ----------------------- This took, 1.126, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(n + 2 k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(n + 2 k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 9 8 7 b(n) = 1/8 (11565805271 n - 131961700340 n + 640549005508 n 6 5 4 - 1728914653136 n + 2848443856247 n - 2961948339624 n 3 2 + 1942512252110 n - 776830448596 n + 172828941120 n - 16443705600) / 2 2 9 8 b(n - 1) / (n %1 (2 n - 1) ) - 1/8 (54989153917 n - 682395727097 n / 7 6 5 + 3651960599461 n - 11030361916253 n + 20648908563060 n 4 3 2 - 24737635408058 n + 18866211068334 n - 8779693495188 n / 2 2 + 2250010065760 n - 241882636800) b(n - 2) / (n %1 (2 n - 1) ) + 1/8 ( / 7 6 5 4 1468481047 n - 13817881286 n + 52081746555 n - 101329547743 n 3 2 2 + 109599843654 n - 65760223043 n + 20251335604 n - 2491924260) (n - 2) / 2 2 5 4 b(n - 3) / (n %1 (2 n - 1) ) - 49/8 (12136207 n - 53516331 n / 3 2 2 2 + 89472959 n - 70214027 n + 25707076 n - 3553340) (n - 2) (n - 3) / 2 2 b(n - 4) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 12136207 n - 114197366 n + 424900353 n - 781092960 n + 709300366 n - 254599940 and in Maple notation b(n) = 1/8*(11565805271*n^9-131961700340*n^8+640549005508*n^7-1728914653136*n^6 +2848443856247*n^5-2961948339624*n^4+1942512252110*n^3-776830448596*n^2+ 172828941120*n-16443705600)/n^2/(12136207*n^5-114197366*n^4+424900353*n^3-\ 781092960*n^2+709300366*n-254599940)/(2*n-1)^2*b(n-1)-1/8*(54989153917*n^9-\ 682395727097*n^8+3651960599461*n^7-11030361916253*n^6+20648908563060*n^5-\ 24737635408058*n^4+18866211068334*n^3-8779693495188*n^2+2250010065760*n-\ 241882636800)/n^2/(12136207*n^5-114197366*n^4+424900353*n^3-781092960*n^2+ 709300366*n-254599940)/(2*n-1)^2*b(n-2)+1/8*(1468481047*n^7-13817881286*n^6+ 52081746555*n^5-101329547743*n^4+109599843654*n^3-65760223043*n^2+20251335604*n -2491924260)*(n-2)^2/n^2/(12136207*n^5-114197366*n^4+424900353*n^3-781092960*n^ 2+709300366*n-254599940)/(2*n-1)^2*b(n-3)-49/8*(12136207*n^5-53516331*n^4+ 89472959*n^3-70214027*n^2+25707076*n-3553340)*(n-2)^2*(n-3)^2/n^2/(12136207*n^5 -114197366*n^4+424900353*n^3-781092960*n^2+709300366*n-254599940)/(2*n-1)^2*b(n -4) Of course, the initial conditions are b(0) = 1, b(1) = 7, b(2) = 93, b(3) = 1519 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 9 8 7 a(n) = 1/8 (11565805271 n - 131961700340 n + 640549005508 n 6 5 4 - 1728914653136 n + 2848443856247 n - 2961948339624 n 3 2 + 1942512252110 n - 776830448596 n + 172828941120 n - 16443705600) / 2 2 9 8 a(n - 1) / (n %1 (2 n - 1) ) - 1/8 (54989153917 n - 682395727097 n / 7 6 5 + 3651960599461 n - 11030361916253 n + 20648908563060 n 4 3 2 - 24737635408058 n + 18866211068334 n - 8779693495188 n / 2 2 + 2250010065760 n - 241882636800) a(n - 2) / (n %1 (2 n - 1) ) + 1/8 ( / 7 6 5 4 1468481047 n - 13817881286 n + 52081746555 n - 101329547743 n 3 2 2 + 109599843654 n - 65760223043 n + 20251335604 n - 2491924260) (n - 2) / 2 2 5 4 a(n - 3) / (n %1 (2 n - 1) ) - 49/8 (12136207 n - 53516331 n / 3 2 2 2 + 89472959 n - 70214027 n + 25707076 n - 3553340) (n - 2) (n - 3) / 2 2 a(n - 4) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 12136207 n - 114197366 n + 424900353 n - 781092960 n + 709300366 n - 254599940 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.34657359027997265470861606072908828403775006718012762706034000474669681098\ 48473578029316634982093438 This constant is identified as, 1/2 ln(2) The implied delta is, -0.7292218309842496641892218664098147725415438432362439\ 332533906650971960984773755589241626043236915564 Since this is negative, there is no Apery-style irrationality proof of, 1/2 ln(2), but still a very fast way to compute it to many digits ----------------------- This took, 1.333, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(n + 2 k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(n + 2 k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 9 8 7 b(n) = 1/12 (76274071861 n - 869390051439 n + 4216366304020 n 6 5 4 - 11372891532002 n + 18730580535525 n - 19477761112693 n 3 2 + 12780066264538 n - 5115343732074 n + 1139367409056 n - 108538191432) / 2 2 9 8 b(n - 1) / (n %1 (2 n - 1) ) - 1/12 (612432689661 n - 7593086463300 n / 7 6 5 + 40594306718901 n - 122469661443116 n + 228962041294173 n 4 3 2 - 273884864454952 n + 208522408116713 n - 96860128482228 n / 2 2 + 24776978186356 n - 2658712114080) b(n - 2) / (n %1 (2 n - 1) ) + 1/12 / 7 6 5 4 (10576279584 n - 99398396448 n + 373878207417 n - 725329159750 n 3 2 + 781846829564 n - 467423857316 n + 143461797727 n - 17591735898) 2 / 2 2 121 5 4 (n - 2) b(n - 3) / (n %1 (2 n - 1) ) - --- (36723193 n - 161517356 n / 12 3 2 2 2 + 269168185 n - 210452554 n + 76796776 n - 10583622) (n - 2) (n - 3) / 2 2 b(n - 4) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 36723193 n - 345133321 n + 1282469539 n - 2354293175 n + 2134891828 n - 765241686 and in Maple notation b(n) = 1/12*(76274071861*n^9-869390051439*n^8+4216366304020*n^7-11372891532002* n^6+18730580535525*n^5-19477761112693*n^4+12780066264538*n^3-5115343732074*n^2+ 1139367409056*n-108538191432)/n^2/(36723193*n^5-345133321*n^4+1282469539*n^3-\ 2354293175*n^2+2134891828*n-765241686)/(2*n-1)^2*b(n-1)-1/12*(612432689661*n^9-\ 7593086463300*n^8+40594306718901*n^7-122469661443116*n^6+228962041294173*n^5-\ 273884864454952*n^4+208522408116713*n^3-96860128482228*n^2+24776978186356*n-\ 2658712114080)/n^2/(36723193*n^5-345133321*n^4+1282469539*n^3-2354293175*n^2+ 2134891828*n-765241686)/(2*n-1)^2*b(n-2)+1/12*(10576279584*n^7-99398396448*n^6+ 373878207417*n^5-725329159750*n^4+781846829564*n^3-467423857316*n^2+ 143461797727*n-17591735898)*(n-2)^2/n^2/(36723193*n^5-345133321*n^4+1282469539* n^3-2354293175*n^2+2134891828*n-765241686)/(2*n-1)^2*b(n-3)-121/12*(36723193*n^ 5-161517356*n^4+269168185*n^3-210452554*n^2+76796776*n-10583622)*(n-2)^2*(n-3)^ 2/n^2/(36723193*n^5-345133321*n^4+1282469539*n^3-2354293175*n^2+2134891828*n-\ 765241686)/(2*n-1)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 10, b(2) = 184, b(3) = 4105 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 9 8 7 a(n) = 1/12 (76274071861 n - 869390051439 n + 4216366304020 n 6 5 4 - 11372891532002 n + 18730580535525 n - 19477761112693 n 3 2 + 12780066264538 n - 5115343732074 n + 1139367409056 n - 108538191432) / 2 2 9 8 a(n - 1) / (n %1 (2 n - 1) ) - 1/12 (612432689661 n - 7593086463300 n / 7 6 5 + 40594306718901 n - 122469661443116 n + 228962041294173 n 4 3 2 - 273884864454952 n + 208522408116713 n - 96860128482228 n / 2 2 + 24776978186356 n - 2658712114080) a(n - 2) / (n %1 (2 n - 1) ) + 1/12 / 7 6 5 4 (10576279584 n - 99398396448 n + 373878207417 n - 725329159750 n 3 2 + 781846829564 n - 467423857316 n + 143461797727 n - 17591735898) 2 / 2 2 121 5 4 (n - 2) a(n - 3) / (n %1 (2 n - 1) ) - --- (36723193 n - 161517356 n / 12 3 2 2 2 + 269168185 n - 210452554 n + 76796776 n - 10583622) (n - 2) (n - 3) / 2 2 a(n - 4) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 36723193 n - 345133321 n + 1282469539 n - 2354293175 n + 2134891828 n - 765241686 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.24413606414846882031005449709389460103277567951616654482993207414166539849\ 30242148608035010697182420 This constant is identified as, 2/9 ln(3) The implied delta is, -0.7949219923570801537590992425684979973495072155357806\ 487317249901233331176410377332074613298722620502 Since this is negative, there is no Apery-style irrationality proof of, 2/9 ln(3), but still a very fast way to compute it to many digits ----------------------- This took, 1.345, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ ) binomial(n, k) binomial(n + k, k) binomial(n + 2 k, k) / ----- k By Shalosh B. Ekhad Let ----- \ b(n) = ) binomial(n, k) binomial(n + k, k) binomial(n + 2 k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 2 (3 n - 2) (30 n - 50 n + 13) b(n - 1) b(n) = -------------------------------------- 2 n (3 n - 4) 3 2 2 (9 n - 30 n + 29 n - 6) b(n - 2) (3 n - 1) (n - 2) b(n - 3) - ---------------------------------- + --------------------------- 2 2 n (3 n - 4) n (3 n - 4) and in Maple notation b(n) = (3*n-2)*(30*n^2-50*n+13)/n^2/(3*n-4)*b(n-1)-(9*n^3-30*n^2+29*n-6)/n^2/(3 *n-4)*b(n-2)+(3*n-1)*(n-2)^2/n^2/(3*n-4)*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 7, b(2) = 115 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 2 (3 n - 2) (30 n - 50 n + 13) a(n - 1) a(n) = -------------------------------------- 2 n (3 n - 4) 3 2 2 (9 n - 30 n + 29 n - 6) a(n - 2) (3 n - 1) (n - 2) a(n - 3) - ---------------------------------- + --------------------------- 2 2 n (3 n - 4) n (3 n - 4) but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.14346964698800134629647774431686847514152356084676321825233536799752040515\ 37106537871634215657399851 The implied delta is, 0.16444191198415579425338050557553390927872515945932884\ 7178334036407146818419916808577331206271324100 Since this is positive, this suggests an Apery-style irrationality proof of, 0.1434696469880013462964777443168684751415235608467632182523353679975204\ 051537106537871634215657399851 ----------------------- This took, 3.015, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + k, k) binomial(n + 2 k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + k, k) binomial(n + 2 k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 2 (3 n - 2) (57 n - 95 n + 25) b(n - 1) b(n) = -------------------------------------- 2 n (3 n - 4) 3 2 2 (9 n - 30 n + 29 n - 6) b(n - 2) (3 n - 1) (n - 2) b(n - 3) - ---------------------------------- + --------------------------- 2 2 n (3 n - 4) n (3 n - 4) and in Maple notation b(n) = (3*n-2)*(57*n^2-95*n+25)/n^2/(3*n-4)*b(n-1)-(9*n^3-30*n^2+29*n-6)/n^2/(3 *n-4)*b(n-2)+(3*n-1)*(n-2)^2/n^2/(3*n-4)*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 13, b(2) = 409 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 2 (3 n - 2) (57 n - 95 n + 25) a(n - 1) a(n) = -------------------------------------- 2 n (3 n - 4) 3 2 2 (9 n - 30 n + 29 n - 6) a(n - 2) (3 n - 1) (n - 2) a(n - 3) - ---------------------------------- + --------------------------- 2 2 n (3 n - 4) n (3 n - 4) but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.07701635339554947882413690238424184089727779270669502823563111216593262466\ 329941284509592522182429862 This constant is identified as, 1/9 ln(2) The implied delta is, 0.20778374377590423771128437544757387878100294839652882\ 5775879219790823183644112529390443490682262721 Since this is positive, this suggests an Apery-style irrationality proof of, 1/9 ln(2) ----------------------- This took, 0.889, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + k, k) binomial(n + 2 k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + k, k) binomial(n + 2 k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 2 (3 n - 2) (84 n - 140 n + 37) b(n - 1) b(n) = --------------------------------------- 2 n (3 n - 4) 3 2 2 (9 n - 30 n + 29 n - 6) b(n - 2) (3 n - 1) (n - 2) b(n - 3) - ---------------------------------- + --------------------------- 2 2 n (3 n - 4) n (3 n - 4) and in Maple notation b(n) = (3*n-2)*(84*n^2-140*n+37)/n^2/(3*n-4)*b(n-1)-(9*n^3-30*n^2+29*n-6)/n^2/( 3*n-4)*b(n-2)+(3*n-1)*(n-2)^2/n^2/(3*n-4)*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 19, b(2) = 883 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 2 (3 n - 2) (84 n - 140 n + 37) a(n - 1) a(n) = --------------------------------------- 2 n (3 n - 4) 3 2 2 (9 n - 30 n + 29 n - 6) a(n - 2) (3 n - 1) (n - 2) a(n - 3) - ---------------------------------- + --------------------------- 2 2 n (3 n - 4) n (3 n - 4) but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.05266121197729926435149028103707346284008790617657377777528917812110456158\ 083182257495374843300288516 The implied delta is, 0.22920459514683997933787709019996781555419456269156642\ 0367269597988852353845153933422933946923027391 Since this is positive, this suggests an Apery-style irrationality proof of, 0.0526612119772992643514902810370734628400879061765737777752891781211045\ 6158083182257495374843300288516 ----------------------- This took, 2.719, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 ) binomial(n, k) binomial(n + 2 k, k) / ----- k By Shalosh B. Ekhad Let ----- \ 2 b(n) = ) binomial(n, k) binomial(n + 2 k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 11 10 9 b(n) = 1/4 (123681111923 n - 2017210594830 n + 14377105139975 n 8 7 6 - 58958186871841 n + 154101689103477 n - 268596297846656 n 5 4 3 + 317263942031347 n - 252823515866929 n + 132593684789878 n 2 / 2 - 43408545306440 n + 7962474352992 n - 620541629856) b(n - 1) / (n %1 / 2 11 10 9 (2 n - 1) ) + 1/4 (59308115757 n - 1026609893727 n + 7780588152803 n 8 7 6 - 33967759267156 n + 94532323852816 n - 175246090304544 n 5 4 3 + 219573652725104 n - 184754573421737 n + 101646676901000 n 2 / 2 - 34654359238716 n + 6588217972160 n - 531414589248) b(n - 2) / (n %1 / 2 10 9 (2 n - 1) ) + 1/4 (n - 2) (43950091291 n - 716815918110 n 8 7 6 + 5015575542441 n - 19714478155965 n + 47906358894012 n 5 4 3 - 74685854643331 n + 75088491491092 n - 47693626626438 n 2 / 2 + 18184819386880 n - 3750002340656 n + 321446641968) b(n - 3) / (n %1 / 2 9 8 (2 n - 1) ) - 1/4 (n - 2) (n - 3) (7679012233 n - 109884905464 n 7 6 5 4 + 650270929995 n - 2075022978680 n + 3914351784267 n - 4504159454802 n 3 2 + 3143336432069 n - 1276230126958 n + 273805532104 n - 24120133776) / 2 2 7 b(n - 4) / (n %1 (2 n - 1) ) + 17/4 (n - 2) (n - 3) (163383239 n / 6 5 4 3 - 1194294039 n + 3485071669 n - 5237300496 n + 4348282118 n 2 2 / 2 - 1983630477 n + 461571452 n - 43039674) (n - 4) b(n - 5) / (n %1 / 2 (2 n - 1) ) 7 6 5 4 %1 := 163383239 n - 2337976712 n + 14081883922 n - 46295482791 n 3 2 + 89752494937 n - 102648455101 n + 64157685996 n - 16916573164 and in Maple notation b(n) = 1/4*(123681111923*n^11-2017210594830*n^10+14377105139975*n^9-\ 58958186871841*n^8+154101689103477*n^7-268596297846656*n^6+317263942031347*n^5-\ 252823515866929*n^4+132593684789878*n^3-43408545306440*n^2+7962474352992*n-\ 620541629856)/n^2/(163383239*n^7-2337976712*n^6+14081883922*n^5-46295482791*n^4 +89752494937*n^3-102648455101*n^2+64157685996*n-16916573164)/(2*n-1)^2*b(n-1)+1 /4*(59308115757*n^11-1026609893727*n^10+7780588152803*n^9-33967759267156*n^8+ 94532323852816*n^7-175246090304544*n^6+219573652725104*n^5-184754573421737*n^4+ 101646676901000*n^3-34654359238716*n^2+6588217972160*n-531414589248)/n^2/( 163383239*n^7-2337976712*n^6+14081883922*n^5-46295482791*n^4+89752494937*n^3-\ 102648455101*n^2+64157685996*n-16916573164)/(2*n-1)^2*b(n-2)+1/4*(n-2)*( 43950091291*n^10-716815918110*n^9+5015575542441*n^8-19714478155965*n^7+ 47906358894012*n^6-74685854643331*n^5+75088491491092*n^4-47693626626438*n^3+ 18184819386880*n^2-3750002340656*n+321446641968)/n^2/(163383239*n^7-2337976712* n^6+14081883922*n^5-46295482791*n^4+89752494937*n^3-102648455101*n^2+ 64157685996*n-16916573164)/(2*n-1)^2*b(n-3)-1/4*(n-2)*(n-3)*(7679012233*n^9-\ 109884905464*n^8+650270929995*n^7-2075022978680*n^6+3914351784267*n^5-\ 4504159454802*n^4+3143336432069*n^3-1276230126958*n^2+273805532104*n-\ 24120133776)/n^2/(163383239*n^7-2337976712*n^6+14081883922*n^5-46295482791*n^4+ 89752494937*n^3-102648455101*n^2+64157685996*n-16916573164)/(2*n-1)^2*b(n-4)+17 /4*(n-2)*(n-3)*(163383239*n^7-1194294039*n^6+3485071669*n^5-5237300496*n^4+ 4348282118*n^3-1983630477*n^2+461571452*n-43039674)*(n-4)^2/n^2/(163383239*n^7-\ 2337976712*n^6+14081883922*n^5-46295482791*n^4+89752494937*n^3-102648455101*n^2 +64157685996*n-16916573164)/(2*n-1)^2*b(n-5) Of course, the initial conditions are b(0) = 1, b(1) = 10, b(2) = 258, b(3) = 8455, b(4) = 307474 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 11 10 9 a(n) = 1/4 (123681111923 n - 2017210594830 n + 14377105139975 n 8 7 6 - 58958186871841 n + 154101689103477 n - 268596297846656 n 5 4 3 + 317263942031347 n - 252823515866929 n + 132593684789878 n 2 / 2 - 43408545306440 n + 7962474352992 n - 620541629856) a(n - 1) / (n %1 / 2 11 10 9 (2 n - 1) ) + 1/4 (59308115757 n - 1026609893727 n + 7780588152803 n 8 7 6 - 33967759267156 n + 94532323852816 n - 175246090304544 n 5 4 3 + 219573652725104 n - 184754573421737 n + 101646676901000 n 2 / 2 - 34654359238716 n + 6588217972160 n - 531414589248) a(n - 2) / (n %1 / 2 10 9 (2 n - 1) ) + 1/4 (n - 2) (43950091291 n - 716815918110 n 8 7 6 + 5015575542441 n - 19714478155965 n + 47906358894012 n 5 4 3 - 74685854643331 n + 75088491491092 n - 47693626626438 n 2 / 2 + 18184819386880 n - 3750002340656 n + 321446641968) a(n - 3) / (n %1 / 2 9 8 (2 n - 1) ) - 1/4 (n - 2) (n - 3) (7679012233 n - 109884905464 n 7 6 5 4 + 650270929995 n - 2075022978680 n + 3914351784267 n - 4504159454802 n 3 2 + 3143336432069 n - 1276230126958 n + 273805532104 n - 24120133776) / 2 2 7 a(n - 4) / (n %1 (2 n - 1) ) + 17/4 (n - 2) (n - 3) (163383239 n / 6 5 4 3 - 1194294039 n + 3485071669 n - 5237300496 n + 4348282118 n 2 2 / 2 - 1983630477 n + 461571452 n - 43039674) (n - 4) a(n - 5) / (n %1 / 2 (2 n - 1) ) 7 6 5 4 %1 := 163383239 n - 2337976712 n + 14081883922 n - 46295482791 n 3 2 + 89752494937 n - 102648455101 n + 64157685996 n - 16916573164 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 0, a(4) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.28557533420667476238184624304999084177492619983329624996709759730998109088\ -3706 19917552667679701630064996 10 This constant is identified as, 0 The implied delta is, -0.2706862828419462291731886216676873278490996141364165\ 957756932536563013012053939499316626707481843025 Since this is negative, there is no Apery-style irrationality proof of, 0, but still a very fast way to compute it to many digits ----------------------- This took, 1.845, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(n + 2 k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(n + 2 k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 11 10 9 b(n) = 1/4 (7101109648980 n - 115898165153910 n + 826684056995274 n 8 7 6 - 3393175170859809 n + 8878319100641770 n - 15494284423684616 n 5 4 3 + 18329474639602944 n - 14633140204896325 n + 7691017256900804 n 2 / - 2524163324698512 n + 464216227124328 n - 36249054940800) b(n - 1) / ( / 2 2 11 10 n %1 (2 n - 1) ) + 1/16 (6810393079458 n - 117963734022369 n 9 8 7 + 894736777583817 n - 3909840712642182 n + 10893627782106698 n 6 5 4 - 20223687561889571 n + 25384527327799827 n - 21407145079369182 n 3 2 + 11810029739965768 n - 4038807375936856 n + 769734877281648 n / 2 2 - 62026137006336) b(n - 2) / (n %1 (2 n - 1) ) + 1/16 (n - 2) ( / 10 9 8 4608572503734 n - 75217131344853 n + 526877859493512 n 7 6 5 - 2074251447539319 n + 5051312717137832 n - 7897662703746809 n 4 3 2 + 7970950453570194 n - 5089543307330263 n + 1954270912823080 n / 2 2 - 406318675302564 n + 34968881060352) b(n - 3) / (n %1 (2 n - 1) ) - / 9 8 1/16 (n - 2) (n - 3) (433691931582 n - 6210960451605 n 7 6 5 + 36793589362398 n - 117579340530177 n + 222262791349007 n 4 3 2 - 256530765570750 n + 179836317144601 n - 73487876370324 n / 2 2 33 + 15886790696124 n - 1403637372288) b(n - 4) / (n %1 (2 n - 1) ) + -- / 16 7 6 5 (n - 2) (n - 3) (4765845402 n - 34891394841 n + 102032978985 n 4 3 2 - 153787437879 n + 128232967739 n - 58857357208 n + 13793592210 n 2 / 2 2 - 1288859472) (n - 4) b(n - 5) / (n %1 (2 n - 1) ) / 7 6 5 4 %1 := 4765845402 n - 68252312655 n + 411464101473 n - 1354127844489 n 3 2 + 2628344994995 n - 3010064353606 n + 1884231143144 n - 497650433736 and in Maple notation b(n) = 1/4*(7101109648980*n^11-115898165153910*n^10+826684056995274*n^9-\ 3393175170859809*n^8+8878319100641770*n^7-15494284423684616*n^6+ 18329474639602944*n^5-14633140204896325*n^4+7691017256900804*n^3-\ 2524163324698512*n^2+464216227124328*n-36249054940800)/n^2/(4765845402*n^7-\ 68252312655*n^6+411464101473*n^5-1354127844489*n^4+2628344994995*n^3-\ 3010064353606*n^2+1884231143144*n-497650433736)/(2*n-1)^2*b(n-1)+1/16*( 6810393079458*n^11-117963734022369*n^10+894736777583817*n^9-3909840712642182*n^ 8+10893627782106698*n^7-20223687561889571*n^6+25384527327799827*n^5-\ 21407145079369182*n^4+11810029739965768*n^3-4038807375936856*n^2+ 769734877281648*n-62026137006336)/n^2/(4765845402*n^7-68252312655*n^6+ 411464101473*n^5-1354127844489*n^4+2628344994995*n^3-3010064353606*n^2+ 1884231143144*n-497650433736)/(2*n-1)^2*b(n-2)+1/16*(n-2)*(4608572503734*n^10-\ 75217131344853*n^9+526877859493512*n^8-2074251447539319*n^7+5051312717137832*n^ 6-7897662703746809*n^5+7970950453570194*n^4-5089543307330263*n^3+ 1954270912823080*n^2-406318675302564*n+34968881060352)/n^2/(4765845402*n^7-\ 68252312655*n^6+411464101473*n^5-1354127844489*n^4+2628344994995*n^3-\ 3010064353606*n^2+1884231143144*n-497650433736)/(2*n-1)^2*b(n-3)-1/16*(n-2)*(n-\ 3)*(433691931582*n^9-6210960451605*n^8+36793589362398*n^7-117579340530177*n^6+ 222262791349007*n^5-256530765570750*n^4+179836317144601*n^3-73487876370324*n^2+ 15886790696124*n-1403637372288)/n^2/(4765845402*n^7-68252312655*n^6+ 411464101473*n^5-1354127844489*n^4+2628344994995*n^3-3010064353606*n^2+ 1884231143144*n-497650433736)/(2*n-1)^2*b(n-4)+33/16*(n-2)*(n-3)*(4765845402*n^ 7-34891394841*n^6+102032978985*n^5-153787437879*n^4+128232967739*n^3-\ 58857357208*n^2+13793592210*n-1288859472)*(n-4)^2/n^2/(4765845402*n^7-\ 68252312655*n^6+411464101473*n^5-1354127844489*n^4+2628344994995*n^3-\ 3010064353606*n^2+1884231143144*n-497650433736)/(2*n-1)^2*b(n-5) Of course, the initial conditions are b(0) = 1, b(1) = 19, b(2) = 965, b(3) = 61891, b(4) = 4400305 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 11 10 9 a(n) = 1/4 (7101109648980 n - 115898165153910 n + 826684056995274 n 8 7 6 - 3393175170859809 n + 8878319100641770 n - 15494284423684616 n 5 4 3 + 18329474639602944 n - 14633140204896325 n + 7691017256900804 n 2 / - 2524163324698512 n + 464216227124328 n - 36249054940800) a(n - 1) / ( / 2 2 11 10 n %1 (2 n - 1) ) + 1/16 (6810393079458 n - 117963734022369 n 9 8 7 + 894736777583817 n - 3909840712642182 n + 10893627782106698 n 6 5 4 - 20223687561889571 n + 25384527327799827 n - 21407145079369182 n 3 2 + 11810029739965768 n - 4038807375936856 n + 769734877281648 n / 2 2 - 62026137006336) a(n - 2) / (n %1 (2 n - 1) ) + 1/16 (n - 2) ( / 10 9 8 4608572503734 n - 75217131344853 n + 526877859493512 n 7 6 5 - 2074251447539319 n + 5051312717137832 n - 7897662703746809 n 4 3 2 + 7970950453570194 n - 5089543307330263 n + 1954270912823080 n / 2 2 - 406318675302564 n + 34968881060352) a(n - 3) / (n %1 (2 n - 1) ) - / 9 8 1/16 (n - 2) (n - 3) (433691931582 n - 6210960451605 n 7 6 5 + 36793589362398 n - 117579340530177 n + 222262791349007 n 4 3 2 - 256530765570750 n + 179836317144601 n - 73487876370324 n / 2 2 33 + 15886790696124 n - 1403637372288) a(n - 4) / (n %1 (2 n - 1) ) + -- / 16 7 6 5 (n - 2) (n - 3) (4765845402 n - 34891394841 n + 102032978985 n 4 3 2 - 153787437879 n + 128232967739 n - 58857357208 n + 13793592210 n 2 / 2 2 - 1288859472) (n - 4) a(n - 5) / (n %1 (2 n - 1) ) / 7 6 5 4 %1 := 4765845402 n - 68252312655 n + 411464101473 n - 1354127844489 n 3 2 + 2628344994995 n - 3010064353606 n + 1884231143144 n - 497650433736 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 0, a(4) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.16240697000248598046154923598729743044354663657316502529893062118482407594\ -4668 12166373340922753931413810 10 This constant is identified as, 0 The implied delta is, -0.2547593663560665096468683088910344053264785189847630\ 598077974825738586088799863547557769675980957566 Since this is negative, there is no Apery-style irrationality proof of, 0, but still a very fast way to compute it to many digits ----------------------- This took, 1.927, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(n + 2 k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(n + 2 k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 11 10 9 b(n) = 1/12 (77949189735857 n - 1272528663118900 n + 9079256199917649 n 8 7 6 - 37278162485415287 n + 97575258854593047 n - 170361415192102422 n 5 4 3 + 201640713546749821 n - 161079335057550359 n + 84724932585199602 n 2 - 27830232664017376 n + 5122776761463792 n - 400284862709472) b(n - 1) / 2 2 11 10 / (n %1 (2 n - 1) ) + 1/36 (37412334422089 n - 648172601047389 n / 9 8 7 + 4917610906352127 n - 21495912433982684 n + 59914818085247328 n 6 5 4 - 111281114133467414 n + 139756902562915284 n - 117939282367848581 n 3 2 + 65117821881275932 n - 22288200272012292 n + 4250453908288960 n / 2 2 - 342353305099008) b(n - 2) / (n %1 (2 n - 1) ) + 1/36 (n - 2) ( / 10 9 8 24539776441389 n - 400614413235300 n + 2807287475335733 n 7 6 5 - 11057951466459789 n + 26948401390127742 n - 42173320760937101 n 4 3 2 + 42617039248355808 n - 27255649274825886 n + 10487769882256684 n / 2 2 - 2185846410630352 n + 188335136898576) b(n - 3) / (n %1 (2 n - 1) ) - / 9 8 1/36 (n - 2) (n - 3) (1579813933995 n - 22630999593510 n 7 6 5 + 134113821251431 n - 428788150258522 n + 811092307255011 n 4 3 2 - 937042876729858 n + 657812466391111 n - 269331488124158 n / 2 2 49 + 58356547336160 n - 5160103814160) b(n - 4) / (n %1 (2 n - 1) ) + -- / 36 7 6 5 (n - 2) (n - 3) (11702325437 n - 85720755967 n + 250851590139 n 4 3 2 - 378449527980 n + 315980239314 n - 145294752333 n + 34121095520 n 2 / 2 2 - 3189843018) (n - 4) b(n - 5) / (n %1 (2 n - 1) ) / 7 6 5 4 %1 := 11702325437 n - 167637034026 n + 1010924960118 n - 3328100208475 n 3 2 + 6462290762259 n - 7404008713227 n + 4636948194604 n - 1225310129708 and in Maple notation b(n) = 1/12*(77949189735857*n^11-1272528663118900*n^10+9079256199917649*n^9-\ 37278162485415287*n^8+97575258854593047*n^7-170361415192102422*n^6+ 201640713546749821*n^5-161079335057550359*n^4+84724932585199602*n^3-\ 27830232664017376*n^2+5122776761463792*n-400284862709472)/n^2/(11702325437*n^7-\ 167637034026*n^6+1010924960118*n^5-3328100208475*n^4+6462290762259*n^3-\ 7404008713227*n^2+4636948194604*n-1225310129708)/(2*n-1)^2*b(n-1)+1/36*( 37412334422089*n^11-648172601047389*n^10+4917610906352127*n^9-21495912433982684 *n^8+59914818085247328*n^7-111281114133467414*n^6+139756902562915284*n^5-\ 117939282367848581*n^4+65117821881275932*n^3-22288200272012292*n^2+ 4250453908288960*n-342353305099008)/n^2/(11702325437*n^7-167637034026*n^6+ 1010924960118*n^5-3328100208475*n^4+6462290762259*n^3-7404008713227*n^2+ 4636948194604*n-1225310129708)/(2*n-1)^2*b(n-2)+1/36*(n-2)*(24539776441389*n^10 -400614413235300*n^9+2807287475335733*n^8-11057951466459789*n^7+ 26948401390127742*n^6-42173320760937101*n^5+42617039248355808*n^4-\ 27255649274825886*n^3+10487769882256684*n^2-2185846410630352*n+188335136898576) /n^2/(11702325437*n^7-167637034026*n^6+1010924960118*n^5-3328100208475*n^4+ 6462290762259*n^3-7404008713227*n^2+4636948194604*n-1225310129708)/(2*n-1)^2*b( n-3)-1/36*(n-2)*(n-3)*(1579813933995*n^9-22630999593510*n^8+134113821251431*n^7 -428788150258522*n^6+811092307255011*n^5-937042876729858*n^4+657812466391111*n^ 3-269331488124158*n^2+58356547336160*n-5160103814160)/n^2/(11702325437*n^7-\ 167637034026*n^6+1010924960118*n^5-3328100208475*n^4+6462290762259*n^3-\ 7404008713227*n^2+4636948194604*n-1225310129708)/(2*n-1)^2*b(n-4)+49/36*(n-2)*( n-3)*(11702325437*n^7-85720755967*n^6+250851590139*n^5-378449527980*n^4+ 315980239314*n^3-145294752333*n^2+34121095520*n-3189843018)*(n-4)^2/n^2/( 11702325437*n^7-167637034026*n^6+1010924960118*n^5-3328100208475*n^4+ 6462290762259*n^3-7404008713227*n^2+4636948194604*n-1225310129708)/(2*n-1)^2*b( n-5) Of course, the initial conditions are b(0) = 1, b(1) = 28, b(2) = 2122, b(3) = 202645, b(4) = 21444994 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 11 10 9 a(n) = 1/12 (77949189735857 n - 1272528663118900 n + 9079256199917649 n 8 7 6 - 37278162485415287 n + 97575258854593047 n - 170361415192102422 n 5 4 3 + 201640713546749821 n - 161079335057550359 n + 84724932585199602 n 2 - 27830232664017376 n + 5122776761463792 n - 400284862709472) a(n - 1) / 2 2 11 10 / (n %1 (2 n - 1) ) + 1/36 (37412334422089 n - 648172601047389 n / 9 8 7 + 4917610906352127 n - 21495912433982684 n + 59914818085247328 n 6 5 4 - 111281114133467414 n + 139756902562915284 n - 117939282367848581 n 3 2 + 65117821881275932 n - 22288200272012292 n + 4250453908288960 n / 2 2 - 342353305099008) a(n - 2) / (n %1 (2 n - 1) ) + 1/36 (n - 2) ( / 10 9 8 24539776441389 n - 400614413235300 n + 2807287475335733 n 7 6 5 - 11057951466459789 n + 26948401390127742 n - 42173320760937101 n 4 3 2 + 42617039248355808 n - 27255649274825886 n + 10487769882256684 n / 2 2 - 2185846410630352 n + 188335136898576) a(n - 3) / (n %1 (2 n - 1) ) - / 9 8 1/36 (n - 2) (n - 3) (1579813933995 n - 22630999593510 n 7 6 5 + 134113821251431 n - 428788150258522 n + 811092307255011 n 4 3 2 - 937042876729858 n + 657812466391111 n - 269331488124158 n / 2 2 49 + 58356547336160 n - 5160103814160) a(n - 4) / (n %1 (2 n - 1) ) + -- / 36 7 6 5 (n - 2) (n - 3) (11702325437 n - 85720755967 n + 250851590139 n 4 3 2 - 378449527980 n + 315980239314 n - 145294752333 n + 34121095520 n 2 / 2 2 - 3189843018) (n - 4) a(n - 5) / (n %1 (2 n - 1) ) / 7 6 5 4 %1 := 11702325437 n - 167637034026 n + 1010924960118 n - 3328100208475 n 3 2 + 6462290762259 n - 7404008713227 n + 4636948194604 n - 1225310129708 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 0, a(4) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.62451491724082803803135135223445760510384512404622471975953700508781759224\ -5230 30274244600412067424645011 10 This constant is identified as, 0 The implied delta is, -0.2482402586583556369688738067825348279745609934807565\ 112772204526639590736086633134453130340892430476 Since this is negative, there is no Apery-style irrationality proof of, 0, but still a very fast way to compute it to many digits ----------------------- This took, 2.182, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 ) binomial(n, k) binomial(2 n, k) / ----- k By Shalosh B. Ekhad Let ----- \ 2 b(n) = ) binomial(n, k) binomial(2 n, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 8 7 6 5 4 b(n) = 1/4 (91390 n - 815480 n + 3029921 n - 6092265 n + 7223517 n 3 2 / 2 - 5167735 n + 2187096 n - 504180 n + 48960) b(n - 1) / (n %1 / 2 (2 n - 1) ) + 2 (2 n - 3) 5 4 3 2 2 (8892 n - 57114 n + 134370 n - 140549 n + 62831 n - 10064) (n - 1) / 2 2 b(n - 2) / (n %1 (2 n - 1) ) + 16 (2 n - 3) (2 n - 5) / 4 3 2 2 / 2 (494 n - 1444 n + 1499 n - 653 n + 102) (n - 2) b(n - 3) / (n %1 / 2 (2 n - 1) ) 4 3 2 %1 := 494 n - 3420 n + 8795 n - 9959 n + 4192 and in Maple notation b(n) = 1/4*(91390*n^8-815480*n^7+3029921*n^6-6092265*n^5+7223517*n^4-5167735*n^ 3+2187096*n^2-504180*n+48960)/n^2/(494*n^4-3420*n^3+8795*n^2-9959*n+4192)/(2*n-\ 1)^2*b(n-1)+2*(2*n-3)*(8892*n^5-57114*n^4+134370*n^3-140549*n^2+62831*n-10064)* (n-1)^2/n^2/(494*n^4-3420*n^3+8795*n^2-9959*n+4192)/(2*n-1)^2*b(n-2)+16*(2*n-3) *(2*n-5)*(494*n^4-1444*n^3+1499*n^2-653*n+102)*(n-2)^2/n^2/(494*n^4-3420*n^3+ 8795*n^2-9959*n+4192)/(2*n-1)^2*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 3, b(2) = 23 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 8 7 6 5 4 a(n) = 1/4 (91390 n - 815480 n + 3029921 n - 6092265 n + 7223517 n 3 2 / 2 - 5167735 n + 2187096 n - 504180 n + 48960) a(n - 1) / (n %1 / 2 (2 n - 1) ) + 2 (2 n - 3) 5 4 3 2 2 (8892 n - 57114 n + 134370 n - 140549 n + 62831 n - 10064) (n - 1) / 2 2 a(n - 2) / (n %1 (2 n - 1) ) + 16 (2 n - 3) (2 n - 5) / 4 3 2 2 / 2 (494 n - 1444 n + 1499 n - 653 n + 102) (n - 2) a(n - 3) / (n %1 / 2 (2 n - 1) ) 4 3 2 %1 := 494 n - 3420 n + 8795 n - 9959 n + 4192 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately -0.1875939585917285211446257548624099188936431880205669423991138362942556400\ -2140 041894338720446045067709965 10 This constant is identified as, 0 The implied delta is, -0.4570032917824090253740724696642378721653913713416813\ 085605005316561844432380597890427375868854996916 Since this is negative, there is no Apery-style irrationality proof of, 0, but still a very fast way to compute it to many digits ----------------------- This took, 1.050, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(2 n, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(2 n, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 8 7 6 5 4 b(n) = (429300 n - 3846528 n + 14331320 n - 28888111 n + 34362820 n 3 2 / 2 - 24699086 n + 10521279 n - 2445237 n + 239598) b(n - 1) / (n %1 / 2 7 6 5 4 (2 n - 1) ) - (2 n - 3) (42400 n - 358704 n + 1287904 n - 2490834 n 3 2 / 2 + 2757199 n - 1730634 n + 570699 n - 74358) b(n - 2) / (n %1 / 2 (2 n - 1) ) + 81 (2 n - 3) (2 n - 5) 4 3 2 2 / 2 (5300 n - 15688 n + 16192 n - 6947 n + 1071) (n - 2) b(n - 3) / (n / 2 %1 (2 n - 1) ) 4 3 2 %1 := 5300 n - 36888 n + 95056 n - 107595 n + 45198 and in Maple notation b(n) = (429300*n^8-3846528*n^7+14331320*n^6-28888111*n^5+34362820*n^4-24699086* n^3+10521279*n^2-2445237*n+239598)/n^2/(5300*n^4-36888*n^3+95056*n^2-107595*n+ 45198)/(2*n-1)^2*b(n-1)-(2*n-3)*(42400*n^7-358704*n^6+1287904*n^5-2490834*n^4+ 2757199*n^3-1730634*n^2+570699*n-74358)/n^2/(5300*n^4-36888*n^3+95056*n^2-\ 107595*n+45198)/(2*n-1)^2*b(n-2)+81*(2*n-3)*(2*n-5)*(5300*n^4-15688*n^3+16192*n ^2-6947*n+1071)*(n-2)^2/n^2/(5300*n^4-36888*n^3+95056*n^2-107595*n+45198)/(2*n-\ 1)^2*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 5, b(2) = 57 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 8 7 6 5 4 a(n) = (429300 n - 3846528 n + 14331320 n - 28888111 n + 34362820 n 3 2 / 2 - 24699086 n + 10521279 n - 2445237 n + 239598) a(n - 1) / (n %1 / 2 7 6 5 4 (2 n - 1) ) - (2 n - 3) (42400 n - 358704 n + 1287904 n - 2490834 n 3 2 / 2 + 2757199 n - 1730634 n + 570699 n - 74358) a(n - 2) / (n %1 / 2 (2 n - 1) ) + 81 (2 n - 3) (2 n - 5) 4 3 2 2 / 2 (5300 n - 15688 n + 16192 n - 6947 n + 1071) (n - 2) a(n - 3) / (n / 2 %1 (2 n - 1) ) 4 3 2 %1 := 5300 n - 36888 n + 95056 n - 107595 n + 45198 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.15403270679109895764827380476848368179455558541339005647126222433186524932\ 65988256901918504436485972 This constant is identified as, 2/9 ln(2) The implied delta is, -0.5365984093536425287473730021928375707968709447135228\ 575206690043799086168464469527846731603384502838 Since this is negative, there is no Apery-style irrationality proof of, 2/9 ln(2), but still a very fast way to compute it to many digits ----------------------- This took, 1.267, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(2 n, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(2 n, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 8 7 6 5 4 b(n) = 1/4 (959760 n - 8637840 n + 32299623 n - 65316510 n + 77940861 n 3 2 / 2 - 56212424 n + 24035422 n - 5609016 n + 552024) b(n - 1) / (n %1 / 2 7 6 5 4 (2 n - 1) ) - 4 (2 n - 3) (74304 n - 631584 n + 2200284 n - 4044870 n 3 2 / 2 + 4208280 n - 2463401 n + 750402 n - 92089) b(n - 2) / (n %1 / 2 (2 n - 1) ) + 256 (2 n - 3) (2 n - 5) 4 3 2 2 / 2 (2064 n - 6192 n + 6423 n - 2754 n + 425) (n - 2) b(n - 3) / (n %1 / 2 (2 n - 1) ) 4 3 2 %1 := 2064 n - 14448 n + 37383 n - 42432 n + 17858 and in Maple notation b(n) = 1/4*(959760*n^8-8637840*n^7+32299623*n^6-65316510*n^5+77940861*n^4-\ 56212424*n^3+24035422*n^2-5609016*n+552024)/n^2/(2064*n^4-14448*n^3+37383*n^2-\ 42432*n+17858)/(2*n-1)^2*b(n-1)-4*(2*n-3)*(74304*n^7-631584*n^6+2200284*n^5-\ 4044870*n^4+4208280*n^3-2463401*n^2+750402*n-92089)/n^2/(2064*n^4-14448*n^3+ 37383*n^2-42432*n+17858)/(2*n-1)^2*b(n-2)+256*(2*n-3)*(2*n-5)*(2064*n^4-6192*n^ 3+6423*n^2-2754*n+425)*(n-2)^2/n^2/(2064*n^4-14448*n^3+37383*n^2-42432*n+17858) /(2*n-1)^2*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 7, b(2) = 103 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 8 7 6 5 4 a(n) = 1/4 (959760 n - 8637840 n + 32299623 n - 65316510 n + 77940861 n 3 2 / 2 - 56212424 n + 24035422 n - 5609016 n + 552024) a(n - 1) / (n %1 / 2 7 6 5 4 (2 n - 1) ) - 4 (2 n - 3) (74304 n - 631584 n + 2200284 n - 4044870 n 3 2 / 2 + 4208280 n - 2463401 n + 750402 n - 92089) a(n - 2) / (n %1 / 2 (2 n - 1) ) + 256 (2 n - 3) (2 n - 5) 4 3 2 2 / 2 (2064 n - 6192 n + 6423 n - 2754 n + 425) (n - 2) a(n - 3) / (n %1 / 2 (2 n - 1) ) 4 3 2 %1 := 2064 n - 14448 n + 37383 n - 42432 n + 17858 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.12924850454918937545826414552029714172323418327326464138055227454558756390\ 80716431616018535074978928 This constant is identified as, 2/17 ln(3) The implied delta is, -0.5909381366514586888753273603886561207140128801021461\ 750056808644950711513441157751501173829712634520 Since this is negative, there is no Apery-style irrationality proof of, 2/17 ln(3), but still a very fast way to compute it to many digits ----------------------- This took, 1.204, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 ) binomial(n, k) binomial(2 n + k, k) / ----- k By Shalosh B. Ekhad Let ----- \ 2 b(n) = ) binomial(n, k) binomial(2 n + k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 2 4 (2 n - 1) b(n - 1) b(n) = --------------------- 2 n and in Maple notation b(n) = 4*(2*n-1)^2/n^2*b(n-1) The recurrence has order less than 2, so we can't do anything. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(2 n + k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(2 n + k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 7 6 5 4 3 b(n) = (140400 n - 1310400 n + 4948350 n - 9707606 n + 10586767 n 2 / 2 - 6385000 n + 1983217 n - 246880) b(n - 1) / ((2 n - 1) %1 n ) - ( / 8 7 6 5 4 975780 n - 11546730 n + 58262796 n - 163248857 n + 276707316 n 3 2 / 2 - 288896678 n + 179907497 n - 60286277 n + 8133280) b(n - 2) / (n / 4 3 2 (2 n - 1) (2 n - 3) %1) + (2340 n - 8970 n + 12188 n - 6801 n + 1264) 2 2 / 2 (n - 2) (2 n - 5) b(n - 3) / (n (2 n - 1) (2 n - 3) %1) / 4 3 2 %1 := 2340 n - 18330 n + 53138 n - 67447 n + 31563 and in Maple notation b(n) = (140400*n^7-1310400*n^6+4948350*n^5-9707606*n^4+10586767*n^3-6385000*n^2 +1983217*n-246880)/(2*n-1)/(2340*n^4-18330*n^3+53138*n^2-67447*n+31563)/n^2*b(n -1)-(975780*n^8-11546730*n^7+58262796*n^6-163248857*n^5+276707316*n^4-288896678 *n^3+179907497*n^2-60286277*n+8133280)/n^2/(2*n-1)/(2*n-3)/(2340*n^4-18330*n^3+ 53138*n^2-67447*n+31563)*b(n-2)+(2340*n^4-8970*n^3+12188*n^2-6801*n+1264)*(n-2) ^2*(2*n-5)^2/n^2/(2*n-1)/(2*n-3)/(2340*n^4-18330*n^3+53138*n^2-67447*n+31563)*b (n-3) Of course, the initial conditions are b(0) = 1, b(1) = 7, b(2) = 101 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 7 6 5 4 3 a(n) = (140400 n - 1310400 n + 4948350 n - 9707606 n + 10586767 n 2 / 2 - 6385000 n + 1983217 n - 246880) a(n - 1) / ((2 n - 1) %1 n ) - ( / 8 7 6 5 4 975780 n - 11546730 n + 58262796 n - 163248857 n + 276707316 n 3 2 / 2 - 288896678 n + 179907497 n - 60286277 n + 8133280) a(n - 2) / (n / 4 3 2 (2 n - 1) (2 n - 3) %1) + (2340 n - 8970 n + 12188 n - 6801 n + 1264) 2 2 / 2 (n - 2) (2 n - 5) a(n - 3) / (n (2 n - 1) (2 n - 3) %1) / 4 3 2 %1 := 2340 n - 18330 n + 53138 n - 67447 n + 31563 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.19804205158855580269063774898805044802157146696007292974876571699811246341\ 99127758873895219989767679 This constant is identified as, 2/7 ln(2) The implied delta is, -0.6430410792864752820422810168512460877923696825863681\ 555340811064424143864025293213811060738615229652 Since this is negative, there is no Apery-style irrationality proof of, 2/7 ln(2), but still a very fast way to compute it to many digits ----------------------- This took, 1.617, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(2 n + k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(2 n + k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 7 6 5 4 3 b(n) = 2 (165213 n - 1541988 n + 5822391 n - 11422678 n + 12462848 n 2 / 2 - 7525190 n + 2341724 n - 292160) b(n - 1) / ((2 n - 1) %1 n ) - 1/4 ( / 8 7 6 5 4 17717670 n - 209659095 n + 1057312692 n - 2959004165 n + 5006016516 n 3 2 / - 5212542530 n + 3234652664 n - 1079283128 n + 144958720) b(n - 2) / ( / 2 n (2 n - 1) (2 n - 3) %1) + 4 3 2 2 2 (3798 n - 14559 n + 19748 n - 10977 n + 2032) (n - 2) (2 n - 5) / 2 b(n - 3) / (n (2 n - 1) (2 n - 3) %1) / 4 3 2 %1 := 3798 n - 29751 n + 86213 n - 109342 n + 51114 and in Maple notation b(n) = 2*(165213*n^7-1541988*n^6+5822391*n^5-11422678*n^4+12462848*n^3-7525190* n^2+2341724*n-292160)/(2*n-1)/(3798*n^4-29751*n^3+86213*n^2-109342*n+51114)/n^2 *b(n-1)-1/4*(17717670*n^8-209659095*n^7+1057312692*n^6-2959004165*n^5+ 5006016516*n^4-5212542530*n^3+3234652664*n^2-1079283128*n+144958720)/n^2/(2*n-1 )/(2*n-3)/(3798*n^4-29751*n^3+86213*n^2-109342*n+51114)*b(n-2)+(3798*n^4-14559* n^3+19748*n^2-10977*n+2032)*(n-2)^2*(2*n-5)^2/n^2/(2*n-1)/(2*n-3)/(3798*n^4-\ 29751*n^3+86213*n^2-109342*n+51114)*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 10, b(2) = 196 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 7 6 5 4 3 a(n) = 2 (165213 n - 1541988 n + 5822391 n - 11422678 n + 12462848 n 2 / 2 - 7525190 n + 2341724 n - 292160) a(n - 1) / ((2 n - 1) %1 n ) - 1/4 ( / 8 7 6 5 4 17717670 n - 209659095 n + 1057312692 n - 2959004165 n + 5006016516 n 3 2 / - 5212542530 n + 3234652664 n - 1079283128 n + 144958720) a(n - 2) / ( / 2 n (2 n - 1) (2 n - 3) %1) + 4 3 2 2 2 (3798 n - 14559 n + 19748 n - 10977 n + 2032) (n - 2) (2 n - 5) / 2 a(n - 3) / (n (2 n - 1) (2 n - 3) %1) / 4 3 2 %1 := 3798 n - 29751 n + 86213 n - 109342 n + 51114 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.15694461266687281305646360527464652923535579397467849310495633337678489903\ 12298524105165364019617270 This constant is identified as, 1/7 ln(3) The implied delta is, -0.7387163603934049784327097940622849376547082179073877\ 090808059543106802428057882712630347919369304769 Since this is negative, there is no Apery-style irrationality proof of, 1/7 ln(3), but still a very fast way to compute it to many digits ----------------------- This took, 1.509, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 ) binomial(n, k) binomial(2 n + 2 k, k) / ----- k By Shalosh B. Ekhad Let ----- \ 2 b(n) = ) binomial(n, k) binomial(2 n + 2 k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 17 16 15 b(n) = 2/9 (23483364611520 n - 633248873215200 n + 7842268620985488 n 14 13 12 - 59176259116732588 n + 304375269420925548 n - 1131055969648436264 n 11 10 + 3139945515837906770 n - 6641753669827708749 n 9 8 + 10820005543539333064 n - 13630552859799932252 n 7 6 + 13256044178949879666 n - 9879674713109900653 n 5 4 3 + 5565183894263052808 n - 2316395259494917614 n + 687169983663160736 n 2 - 136767051939113880 n + 16293519136641600 n - 874779810912000) b(n - 1) / 2 2 2 17 / (n %1 (3 n - 1) (3 n - 2) ) - 4/9 (27817938441120 n / 16 15 14 - 777952276702320 n + 10030774054856688 n - 79141267090472332 n 13 12 + 427587074052228988 n - 1677251569760774892 n 11 10 + 4940837921091813374 n - 11150312161917911393 n 9 8 + 19487851717711047966 n - 26482263332780424296 n 7 6 + 27924525365091408845 n - 22665508662551929176 n 5 4 + 13949866511526698263 n - 6354066664650926851 n 3 2 + 2061398549040087156 n - 447186995298663540 n + 57710281016102400 n / 2 2 2 - 3327420122208000) b(n - 2) / (n %1 (3 n - 1) (3 n - 2) ) - 4/9 ( / 15 14 13 3031252991040 n - 75677805628320 n + 858344525581776 n 12 11 10 - 5857769424107608 n + 26854371245246884 n - 87440281467938624 n 9 8 7 + 208486612373084912 n - 369859713845792208 n + 491028173462289951 n 6 5 4 - 486556397579760371 n + 355858012913663087 n - 188041153939272139 n 3 2 + 69248122603406110 n - 16717449784653690 n + 2358040920379200 n 2 / 2 2 2 - 146268798354000) (n - 2) b(n - 3) / (n %1 (3 n - 1) (3 n - 2) ) - 4 / 11 10 9 8 (2948689680 n - 38232266280 n + 217211366892 n - 712740293240 n 7 6 5 + 1498419303272 n - 2115073845538 n + 2040519733965 n 4 3 2 - 1341682903844 n + 587361445501 n - 162521972848 n + 25510883640 n 2 2 2 / 2 2 - 1720270800) (2 n - 7) (n - 2) (n - 3) b(n - 4) / (n %1 (3 n - 1) / 2 (3 n - 2) ) 11 10 9 8 %1 := 2948689680 n - 70667852760 n + 761711962092 n - 4874628375068 n 7 6 5 + 20580890405304 n - 60197562549050 n + 124476682560457 n 4 3 2 - 181977361139651 n + 184339059916975 n - 123231781457679 n + 48932930544400 n - 8743942975500 and in Maple notation b(n) = 2/9*(23483364611520*n^17-633248873215200*n^16+7842268620985488*n^15-\ 59176259116732588*n^14+304375269420925548*n^13-1131055969648436264*n^12+ 3139945515837906770*n^11-6641753669827708749*n^10+10820005543539333064*n^9-\ 13630552859799932252*n^8+13256044178949879666*n^7-9879674713109900653*n^6+ 5565183894263052808*n^5-2316395259494917614*n^4+687169983663160736*n^3-\ 136767051939113880*n^2+16293519136641600*n-874779810912000)/n^2/(2948689680*n^ 11-70667852760*n^10+761711962092*n^9-4874628375068*n^8+20580890405304*n^7-\ 60197562549050*n^6+124476682560457*n^5-181977361139651*n^4+184339059916975*n^3-\ 123231781457679*n^2+48932930544400*n-8743942975500)/(3*n-1)^2/(3*n-2)^2*b(n-1)-\ 4/9*(27817938441120*n^17-777952276702320*n^16+10030774054856688*n^15-\ 79141267090472332*n^14+427587074052228988*n^13-1677251569760774892*n^12+ 4940837921091813374*n^11-11150312161917911393*n^10+19487851717711047966*n^9-\ 26482263332780424296*n^8+27924525365091408845*n^7-22665508662551929176*n^6+ 13949866511526698263*n^5-6354066664650926851*n^4+2061398549040087156*n^3-\ 447186995298663540*n^2+57710281016102400*n-3327420122208000)/n^2/(2948689680*n^ 11-70667852760*n^10+761711962092*n^9-4874628375068*n^8+20580890405304*n^7-\ 60197562549050*n^6+124476682560457*n^5-181977361139651*n^4+184339059916975*n^3-\ 123231781457679*n^2+48932930544400*n-8743942975500)/(3*n-1)^2/(3*n-2)^2*b(n-2)-\ 4/9*(3031252991040*n^15-75677805628320*n^14+858344525581776*n^13-\ 5857769424107608*n^12+26854371245246884*n^11-87440281467938624*n^10+ 208486612373084912*n^9-369859713845792208*n^8+491028173462289951*n^7-\ 486556397579760371*n^6+355858012913663087*n^5-188041153939272139*n^4+ 69248122603406110*n^3-16717449784653690*n^2+2358040920379200*n-146268798354000) *(n-2)^2/n^2/(2948689680*n^11-70667852760*n^10+761711962092*n^9-4874628375068*n ^8+20580890405304*n^7-60197562549050*n^6+124476682560457*n^5-181977361139651*n^ 4+184339059916975*n^3-123231781457679*n^2+48932930544400*n-8743942975500)/(3*n-\ 1)^2/(3*n-2)^2*b(n-3)-4*(2948689680*n^11-38232266280*n^10+217211366892*n^9-\ 712740293240*n^8+1498419303272*n^7-2115073845538*n^6+2040519733965*n^5-\ 1341682903844*n^4+587361445501*n^3-162521972848*n^2+25510883640*n-1720270800)*( 2*n-7)^2*(n-2)^2*(n-3)^2/n^2/(2948689680*n^11-70667852760*n^10+761711962092*n^9 -4874628375068*n^8+20580890405304*n^7-60197562549050*n^6+124476682560457*n^5-\ 181977361139651*n^4+184339059916975*n^3-123231781457679*n^2+48932930544400*n-\ 8743942975500)/(3*n-1)^2/(3*n-2)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 5, b(2) = 53, b(3) = 698 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 17 16 15 a(n) = 2/9 (23483364611520 n - 633248873215200 n + 7842268620985488 n 14 13 12 - 59176259116732588 n + 304375269420925548 n - 1131055969648436264 n 11 10 + 3139945515837906770 n - 6641753669827708749 n 9 8 + 10820005543539333064 n - 13630552859799932252 n 7 6 + 13256044178949879666 n - 9879674713109900653 n 5 4 3 + 5565183894263052808 n - 2316395259494917614 n + 687169983663160736 n 2 - 136767051939113880 n + 16293519136641600 n - 874779810912000) a(n - 1) / 2 2 2 17 / (n %1 (3 n - 1) (3 n - 2) ) - 4/9 (27817938441120 n / 16 15 14 - 777952276702320 n + 10030774054856688 n - 79141267090472332 n 13 12 + 427587074052228988 n - 1677251569760774892 n 11 10 + 4940837921091813374 n - 11150312161917911393 n 9 8 + 19487851717711047966 n - 26482263332780424296 n 7 6 + 27924525365091408845 n - 22665508662551929176 n 5 4 + 13949866511526698263 n - 6354066664650926851 n 3 2 + 2061398549040087156 n - 447186995298663540 n + 57710281016102400 n / 2 2 2 - 3327420122208000) a(n - 2) / (n %1 (3 n - 1) (3 n - 2) ) - 4/9 ( / 15 14 13 3031252991040 n - 75677805628320 n + 858344525581776 n 12 11 10 - 5857769424107608 n + 26854371245246884 n - 87440281467938624 n 9 8 7 + 208486612373084912 n - 369859713845792208 n + 491028173462289951 n 6 5 4 - 486556397579760371 n + 355858012913663087 n - 188041153939272139 n 3 2 + 69248122603406110 n - 16717449784653690 n + 2358040920379200 n 2 / 2 2 2 - 146268798354000) (n - 2) a(n - 3) / (n %1 (3 n - 1) (3 n - 2) ) - 4 / 11 10 9 8 (2948689680 n - 38232266280 n + 217211366892 n - 712740293240 n 7 6 5 + 1498419303272 n - 2115073845538 n + 2040519733965 n 4 3 2 - 1341682903844 n + 587361445501 n - 162521972848 n + 25510883640 n 2 2 2 / 2 2 - 1720270800) (2 n - 7) (n - 2) (n - 3) a(n - 4) / (n %1 (3 n - 1) / 2 (3 n - 2) ) 11 10 9 8 %1 := 2948689680 n - 70667852760 n + 761711962092 n - 4874628375068 n 7 6 5 + 20580890405304 n - 60197562549050 n + 124476682560457 n 4 3 2 - 181977361139651 n + 184339059916975 n - 123231781457679 n + 48932930544400 n - 8743942975500 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.59352100953272899805385962026306246672174646576686509838857397393978740551\ -1660 37451810209355403265836566 10 This constant is identified as, 0 The implied delta is, -0.6783578054091546747679095772754604999740396339229488\ 789130475779848087657110039663512460509477768583 Since this is negative, there is no Apery-style irrationality proof of, 0, but still a very fast way to compute it to many digits ----------------------- This took, 1.973, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(2 n + 2 k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(2 n + 2 k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 17 16 b(n) = 1/9 (18092532751141376 n - 486618153132971264 n 15 14 + 6010946479766827360 n - 45244349751504944000 n 13 12 + 232159998910923384088 n - 860769344658308017300 n 11 10 + 2384688910394248360004 n - 5035016691268019735592 n 9 8 + 8189837229356521345403 n - 10304594894886792723505 n 7 6 + 10012861734167487391394 n - 7459065466484449635158 n 5 4 + 4201427468669811049943 n - 1749371705889604329241 n 3 2 + 519336681564968787552 n - 103471023106252433460 n / 2 + 12342495716358796800 n - 663571233932544000) b(n - 1) / (n %1 / 2 2 17 (3 n - 1) (3 n - 2) ) - 1/9 (124922084678017792 n 16 15 - 3484836087193930880 n + 44815314175826729344 n 14 13 - 352616533341599963392 n + 1899650862197472616224 n 12 11 - 7429078372706530876920 n + 21815162558358065717552 n 10 9 - 49067647151442554734690 n + 85457101738370608550925 n 8 7 - 115701518017820633864504 n + 121533131882388912640983 n 6 5 - 98249644165570269276843 n + 60219590570355955405104 n 4 3 - 27314348514556139543031 n + 8824208833686845753756 n 2 - 1906446543992189648460 n + 245076970005096243840 n / 2 2 2 - 14079459260568844800) b(n - 2) / (n %1 (3 n - 1) (3 n - 2) ) - 1/9 ( / 15 14 13 7245276750557824 n - 180378967521101888 n + 2039419699026827552 n 12 11 - 13868843806419006608 n + 63330968461145913040 n 10 9 - 205321394276210699616 n + 487252205789598491980 n 8 7 - 860016324398529127738 n + 1135609198990487603347 n 6 5 - 1118909747089226696593 n + 813587138475014624093 n 4 3 - 427389877727891126799 n + 156482716684820503684 n 2 - 37568998970058448998 n + 5272190222361793920 n - 325511525123265600) 2 / 2 2 2 (n - 2) b(n - 3) / (n %1 (3 n - 1) (3 n - 2) ) - 49/9 ( / 11 10 9 607621330976 n - 7835931442000 n + 44258152291232 n 8 7 6 - 144319334255664 n + 301434372901332 n - 422662953495638 n 5 4 3 + 405067470679503 n - 264626965553536 n + 115143604431207 n 2 2 2 - 31682222097812 n + 4948430691680 n - 332219630400) (2 n - 7) (n - 2) 2 / 2 2 2 (n - 3) b(n - 4) / (n %1 (3 n - 1) (3 n - 2) ) / 11 10 9 %1 := 607621330976 n - 14519766082736 n + 156036639914912 n 8 7 6 - 995517139377792 n + 4190109341693076 n - 12217596373157954 n 5 4 3 + 25184952707888631 n - 36705052901566033 n + 37067925098331473 n 2 - 24705886598606845 n + 9781528428802872 n - 1742919278800980 and in Maple notation b(n) = 1/9*(18092532751141376*n^17-486618153132971264*n^16+6010946479766827360* n^15-45244349751504944000*n^14+232159998910923384088*n^13-860769344658308017300 *n^12+2384688910394248360004*n^11-5035016691268019735592*n^10+ 8189837229356521345403*n^9-10304594894886792723505*n^8+10012861734167487391394* n^7-7459065466484449635158*n^6+4201427468669811049943*n^5-\ 1749371705889604329241*n^4+519336681564968787552*n^3-103471023106252433460*n^2+ 12342495716358796800*n-663571233932544000)/n^2/(607621330976*n^11-\ 14519766082736*n^10+156036639914912*n^9-995517139377792*n^8+4190109341693076*n^ 7-12217596373157954*n^6+25184952707888631*n^5-36705052901566033*n^4+ 37067925098331473*n^3-24705886598606845*n^2+9781528428802872*n-1742919278800980 )/(3*n-1)^2/(3*n-2)^2*b(n-1)-1/9*(124922084678017792*n^17-3484836087193930880*n ^16+44815314175826729344*n^15-352616533341599963392*n^14+1899650862197472616224 *n^13-7429078372706530876920*n^12+21815162558358065717552*n^11-\ 49067647151442554734690*n^10+85457101738370608550925*n^9-\ 115701518017820633864504*n^8+121533131882388912640983*n^7-\ 98249644165570269276843*n^6+60219590570355955405104*n^5-27314348514556139543031 *n^4+8824208833686845753756*n^3-1906446543992189648460*n^2+ 245076970005096243840*n-14079459260568844800)/n^2/(607621330976*n^11-\ 14519766082736*n^10+156036639914912*n^9-995517139377792*n^8+4190109341693076*n^ 7-12217596373157954*n^6+25184952707888631*n^5-36705052901566033*n^4+ 37067925098331473*n^3-24705886598606845*n^2+9781528428802872*n-1742919278800980 )/(3*n-1)^2/(3*n-2)^2*b(n-2)-1/9*(7245276750557824*n^15-180378967521101888*n^14 +2039419699026827552*n^13-13868843806419006608*n^12+63330968461145913040*n^11-\ 205321394276210699616*n^10+487252205789598491980*n^9-860016324398529127738*n^8+ 1135609198990487603347*n^7-1118909747089226696593*n^6+813587138475014624093*n^5 -427389877727891126799*n^4+156482716684820503684*n^3-37568998970058448998*n^2+ 5272190222361793920*n-325511525123265600)*(n-2)^2/n^2/(607621330976*n^11-\ 14519766082736*n^10+156036639914912*n^9-995517139377792*n^8+4190109341693076*n^ 7-12217596373157954*n^6+25184952707888631*n^5-36705052901566033*n^4+ 37067925098331473*n^3-24705886598606845*n^2+9781528428802872*n-1742919278800980 )/(3*n-1)^2/(3*n-2)^2*b(n-3)-49/9*(607621330976*n^11-7835931442000*n^10+ 44258152291232*n^9-144319334255664*n^8+301434372901332*n^7-422662953495638*n^6+ 405067470679503*n^5-264626965553536*n^4+115143604431207*n^3-31682222097812*n^2+ 4948430691680*n-332219630400)*(2*n-7)^2*(n-2)^2*(n-3)^2/n^2/(607621330976*n^11-\ 14519766082736*n^10+156036639914912*n^9-995517139377792*n^8+4190109341693076*n^ 7-12217596373157954*n^6+25184952707888631*n^5-36705052901566033*n^4+ 37067925098331473*n^3-24705886598606845*n^2+9781528428802872*n-1742919278800980 )/(3*n-1)^2/(3*n-2)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 9, b(2) = 161, b(3) = 3525 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 17 16 a(n) = 1/9 (18092532751141376 n - 486618153132971264 n 15 14 + 6010946479766827360 n - 45244349751504944000 n 13 12 + 232159998910923384088 n - 860769344658308017300 n 11 10 + 2384688910394248360004 n - 5035016691268019735592 n 9 8 + 8189837229356521345403 n - 10304594894886792723505 n 7 6 + 10012861734167487391394 n - 7459065466484449635158 n 5 4 + 4201427468669811049943 n - 1749371705889604329241 n 3 2 + 519336681564968787552 n - 103471023106252433460 n / 2 + 12342495716358796800 n - 663571233932544000) a(n - 1) / (n %1 / 2 2 17 (3 n - 1) (3 n - 2) ) - 1/9 (124922084678017792 n 16 15 - 3484836087193930880 n + 44815314175826729344 n 14 13 - 352616533341599963392 n + 1899650862197472616224 n 12 11 - 7429078372706530876920 n + 21815162558358065717552 n 10 9 - 49067647151442554734690 n + 85457101738370608550925 n 8 7 - 115701518017820633864504 n + 121533131882388912640983 n 6 5 - 98249644165570269276843 n + 60219590570355955405104 n 4 3 - 27314348514556139543031 n + 8824208833686845753756 n 2 - 1906446543992189648460 n + 245076970005096243840 n / 2 2 2 - 14079459260568844800) a(n - 2) / (n %1 (3 n - 1) (3 n - 2) ) - 1/9 ( / 15 14 13 7245276750557824 n - 180378967521101888 n + 2039419699026827552 n 12 11 - 13868843806419006608 n + 63330968461145913040 n 10 9 - 205321394276210699616 n + 487252205789598491980 n 8 7 - 860016324398529127738 n + 1135609198990487603347 n 6 5 - 1118909747089226696593 n + 813587138475014624093 n 4 3 - 427389877727891126799 n + 156482716684820503684 n 2 - 37568998970058448998 n + 5272190222361793920 n - 325511525123265600) 2 / 2 2 2 (n - 2) a(n - 3) / (n %1 (3 n - 1) (3 n - 2) ) - 49/9 ( / 11 10 9 607621330976 n - 7835931442000 n + 44258152291232 n 8 7 6 - 144319334255664 n + 301434372901332 n - 422662953495638 n 5 4 3 + 405067470679503 n - 264626965553536 n + 115143604431207 n 2 2 2 - 31682222097812 n + 4948430691680 n - 332219630400) (2 n - 7) (n - 2) 2 / 2 2 2 (n - 3) a(n - 4) / (n %1 (3 n - 1) (3 n - 2) ) / 11 10 9 %1 := 607621330976 n - 14519766082736 n + 156036639914912 n 8 7 6 - 995517139377792 n + 4190109341693076 n - 12217596373157954 n 5 4 3 + 25184952707888631 n - 36705052901566033 n + 37067925098331473 n 2 - 24705886598606845 n + 9781528428802872 n - 1742919278800980 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.37808028030542471422758115715900540076845461874195741133855273245094197561\ 98334812395618147253192841 This constant is identified as, 6/11 ln(2) The implied delta is, -0.8013547722523136255126419149108569649659235985034022\ 581943453134271623041819783074145919492179874367 Since this is negative, there is no Apery-style irrationality proof of, 6/11 ln(2), but still a very fast way to compute it to many digits ----------------------- This took, 2.227, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(2 n + 2 k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(2 n + 2 k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 17 16 b(n) = 2/81 (2358935725899194304 n - 63394844959235863008 n 15 14 + 782460690934603085808 n - 5885024805565804833540 n 13 12 + 30175284220832370829276 n - 111802725921905799268700 n 11 10 + 309547068388448746161950 n - 653219804526804075763057 n 9 8 + 1062032263224920890139112 n - 1335808439758219420162320 n 7 6 + 1297701389096143298252942 n - 966631963658169724578001 n 5 4 + 544493879405538788832400 n - 226754586475544485248902 n 3 2 + 67337119988235867674496 n - 13421478190703416356600 n / 2 + 1601739167805021268800 n - 86158499011976160000) b(n - 1) / (n %1 / 2 2 17 (3 n - 1) (3 n - 2) ) - 4/81 (13646809316385760032 n 16 15 - 380395827232400485296 n + 4887949692980678853408 n 14 13 - 38426854998918670199964 n + 206834119838171887259724 n 12 11 - 808133299835978188541780 n + 2370765402540006181899322 n 10 9 - 5327074083933693413671873 n + 9267987333997677533512646 n 8 7 - 12534298449572909320036320 n + 13151028076735751708326749 n 6 5 - 10618964802103880966214604 n + 6500715249358826615292851 n 4 3 - 2944955964470072156536751 n + 950233523607024950713444 n 2 - 205050098157330957488964 n + 26329672929506336603136 n / 2 2 2 - 1511017193706858351360) b(n - 2) / (n %1 (3 n - 1) (3 n - 2) ) - / 15 14 4/81 (540486261895045824 n - 13444240011355346400 n 13 12 + 151855145579191827696 n - 1031538161531552781384 n 11 10 + 4704694718690048766780 n - 15232376040260007402608 n 9 8 + 36095625570676791867124 n - 63610288263418322878104 n 7 6 + 83854826334512819596323 n - 82478417235644316120063 n 5 4 + 59865349693729833075807 n - 31391938373792343651103 n 3 2 + 11473568065322055472622 n - 2750026809996291428082 n 2 / + 385327222956808126848 n - 23757340836022350480) (n - 2) b(n - 3) / ( / 2 2 2 484 11 n %1 (3 n - 1) (3 n - 2) ) - --- (12020422157616 n 81 10 9 8 - 154755023321208 n + 872471515410876 n - 2839448512077168 n 7 6 5 + 5918568585524432 n - 8281583070080114 n + 7920320642425985 n 4 3 2 - 5163792420885092 n + 2242539031453145 n - 615955265478560 n 2 2 2 + 96054730089624 n - 6439837821840) (2 n - 7) (n - 2) (n - 3) b(n - 4) / 2 2 2 / (n %1 (3 n - 1) (3 n - 2) ) / 11 10 9 %1 := 12020422157616 n - 286979667054984 n + 3081144967291836 n 8 7 6 - 19639037856236052 n + 82580473347491552 n - 240555718735697698 n 5 4 3 + 495391987890774533 n - 721297143174210943 n + 727733173459597355 n 2 - 484581851817698659 n + 191679440382489264 n - 34123949056725660 and in Maple notation b(n) = 2/81*(2358935725899194304*n^17-63394844959235863008*n^16+ 782460690934603085808*n^15-5885024805565804833540*n^14+30175284220832370829276* n^13-111802725921905799268700*n^12+309547068388448746161950*n^11-\ 653219804526804075763057*n^10+1062032263224920890139112*n^9-\ 1335808439758219420162320*n^8+1297701389096143298252942*n^7-\ 966631963658169724578001*n^6+544493879405538788832400*n^5-\ 226754586475544485248902*n^4+67337119988235867674496*n^3-\ 13421478190703416356600*n^2+1601739167805021268800*n-86158499011976160000)/n^2/ (12020422157616*n^11-286979667054984*n^10+3081144967291836*n^9-\ 19639037856236052*n^8+82580473347491552*n^7-240555718735697698*n^6+ 495391987890774533*n^5-721297143174210943*n^4+727733173459597355*n^3-\ 484581851817698659*n^2+191679440382489264*n-34123949056725660)/(3*n-1)^2/(3*n-2 )^2*b(n-1)-4/81*(13646809316385760032*n^17-380395827232400485296*n^16+ 4887949692980678853408*n^15-38426854998918670199964*n^14+ 206834119838171887259724*n^13-808133299835978188541780*n^12+ 2370765402540006181899322*n^11-5327074083933693413671873*n^10+ 9267987333997677533512646*n^9-12534298449572909320036320*n^8+ 13151028076735751708326749*n^7-10618964802103880966214604*n^6+ 6500715249358826615292851*n^5-2944955964470072156536751*n^4+ 950233523607024950713444*n^3-205050098157330957488964*n^2+ 26329672929506336603136*n-1511017193706858351360)/n^2/(12020422157616*n^11-\ 286979667054984*n^10+3081144967291836*n^9-19639037856236052*n^8+ 82580473347491552*n^7-240555718735697698*n^6+495391987890774533*n^5-\ 721297143174210943*n^4+727733173459597355*n^3-484581851817698659*n^2+ 191679440382489264*n-34123949056725660)/(3*n-1)^2/(3*n-2)^2*b(n-2)-4/81*( 540486261895045824*n^15-13444240011355346400*n^14+151855145579191827696*n^13-\ 1031538161531552781384*n^12+4704694718690048766780*n^11-15232376040260007402608 *n^10+36095625570676791867124*n^9-63610288263418322878104*n^8+ 83854826334512819596323*n^7-82478417235644316120063*n^6+59865349693729833075807 *n^5-31391938373792343651103*n^4+11473568065322055472622*n^3-\ 2750026809996291428082*n^2+385327222956808126848*n-23757340836022350480)*(n-2)^ 2/n^2/(12020422157616*n^11-286979667054984*n^10+3081144967291836*n^9-\ 19639037856236052*n^8+82580473347491552*n^7-240555718735697698*n^6+ 495391987890774533*n^5-721297143174210943*n^4+727733173459597355*n^3-\ 484581851817698659*n^2+191679440382489264*n-34123949056725660)/(3*n-1)^2/(3*n-2 )^2*b(n-3)-484/81*(12020422157616*n^11-154755023321208*n^10+872471515410876*n^9 -2839448512077168*n^8+5918568585524432*n^7-8281583070080114*n^6+ 7920320642425985*n^5-5163792420885092*n^4+2242539031453145*n^3-615955265478560* n^2+96054730089624*n-6439837821840)*(2*n-7)^2*(n-2)^2*(n-3)^2/n^2/( 12020422157616*n^11-286979667054984*n^10+3081144967291836*n^9-19639037856236052 *n^8+82580473347491552*n^7-240555718735697698*n^6+495391987890774533*n^5-\ 721297143174210943*n^4+727733173459597355*n^3-484581851817698659*n^2+ 191679440382489264*n-34123949056725660)/(3*n-1)^2/(3*n-2)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 13, b(2) = 325, b(3) = 9802 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 17 16 a(n) = 2/81 (2358935725899194304 n - 63394844959235863008 n 15 14 + 782460690934603085808 n - 5885024805565804833540 n 13 12 + 30175284220832370829276 n - 111802725921905799268700 n 11 10 + 309547068388448746161950 n - 653219804526804075763057 n 9 8 + 1062032263224920890139112 n - 1335808439758219420162320 n 7 6 + 1297701389096143298252942 n - 966631963658169724578001 n 5 4 + 544493879405538788832400 n - 226754586475544485248902 n 3 2 + 67337119988235867674496 n - 13421478190703416356600 n / 2 + 1601739167805021268800 n - 86158499011976160000) a(n - 1) / (n %1 / 2 2 17 (3 n - 1) (3 n - 2) ) - 4/81 (13646809316385760032 n 16 15 - 380395827232400485296 n + 4887949692980678853408 n 14 13 - 38426854998918670199964 n + 206834119838171887259724 n 12 11 - 808133299835978188541780 n + 2370765402540006181899322 n 10 9 - 5327074083933693413671873 n + 9267987333997677533512646 n 8 7 - 12534298449572909320036320 n + 13151028076735751708326749 n 6 5 - 10618964802103880966214604 n + 6500715249358826615292851 n 4 3 - 2944955964470072156536751 n + 950233523607024950713444 n 2 - 205050098157330957488964 n + 26329672929506336603136 n / 2 2 2 - 1511017193706858351360) a(n - 2) / (n %1 (3 n - 1) (3 n - 2) ) - / 15 14 4/81 (540486261895045824 n - 13444240011355346400 n 13 12 + 151855145579191827696 n - 1031538161531552781384 n 11 10 + 4704694718690048766780 n - 15232376040260007402608 n 9 8 + 36095625570676791867124 n - 63610288263418322878104 n 7 6 + 83854826334512819596323 n - 82478417235644316120063 n 5 4 + 59865349693729833075807 n - 31391938373792343651103 n 3 2 + 11473568065322055472622 n - 2750026809996291428082 n 2 / + 385327222956808126848 n - 23757340836022350480) (n - 2) a(n - 3) / ( / 2 2 2 484 11 n %1 (3 n - 1) (3 n - 2) ) - --- (12020422157616 n 81 10 9 8 - 154755023321208 n + 872471515410876 n - 2839448512077168 n 7 6 5 + 5918568585524432 n - 8281583070080114 n + 7920320642425985 n 4 3 2 - 5163792420885092 n + 2242539031453145 n - 615955265478560 n 2 2 2 + 96054730089624 n - 6439837821840) (2 n - 7) (n - 2) (n - 3) a(n - 4) / 2 2 2 / (n %1 (3 n - 1) (3 n - 2) ) / 11 10 9 %1 := 12020422157616 n - 286979667054984 n + 3081144967291836 n 8 7 6 - 19639037856236052 n + 82580473347491552 n - 240555718735697698 n 5 4 3 + 495391987890774533 n - 721297143174210943 n + 727733173459597355 n 2 - 484581851817698659 n + 191679440382489264 n - 34123949056725660 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.24413606414846882031005449709389460103277567951616654482993207414166539849\ 30242148608035010697182420 This constant is identified as, 2/9 ln(3) The implied delta is, -0.8480636789246993798140694984956357425546943949709526\ 251021911038449458544918368071979112958215773555 Since this is negative, there is no Apery-style irrationality proof of, 2/9 ln(3), but still a very fast way to compute it to many digits ----------------------- This took, 2.706, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ ) binomial(n, k) binomial(n + k, k) binomial(2 n, k) / ----- k By Shalosh B. Ekhad Let ----- \ b(n) = ) binomial(n, k) binomial(n + k, k) binomial(2 n, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients b(n) = 1/4 6 5 4 3 2 (29665 n - 141345 n + 264772 n - 249181 n + 124975 n - 31902 n + 3276) / 2 2 2 b(n - 1) / (n (85 n - 235 n + 163) (2 n - 1) ) / 2 2 2 (85 n - 65 n + 13) (n - 1) (2 n - 3) b(n - 2) + ------------------------------------------------ 2 2 2 n (85 n - 235 n + 163) (2 n - 1) and in Maple notation b(n) = 1/4*(29665*n^6-141345*n^5+264772*n^4-249181*n^3+124975*n^2-31902*n+3276) /n^2/(85*n^2-235*n+163)/(2*n-1)^2*b(n-1)+(85*n^2-65*n+13)*(n-1)^2*(2*n-3)^2/n^2 /(85*n^2-235*n+163)/(2*n-1)^2*b(n-2) Of course, the initial conditions are b(0) = 1, b(1) = 5 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. a(n) = 1/4 6 5 4 3 2 (29665 n - 141345 n + 264772 n - 249181 n + 124975 n - 31902 n + 3276) / 2 2 2 a(n - 1) / (n (85 n - 235 n + 163) (2 n - 1) ) / 2 2 2 (85 n - 65 n + 13) (n - 1) (2 n - 3) a(n - 2) + ------------------------------------------------ 2 2 2 n (85 n - 235 n + 163) (2 n - 1) but with the following simpler initial conditions a(0) = 0, a(1) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.19938594749675471957241395959345759869320604863112708336188584598424332974\ 58425301616519879539101461 2 2 Pi This constant is identified as, ----- 99 The implied delta is, -0.1258541371681989545806168285292593001026961904084460\ 573649849848302616815941689599194050953982940120 2 2 Pi Since this is negative, there is no Apery-style irrationality proof of, -----, 99 but still a very fast way to compute it to many digits ----------------------- This took, 5.584, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + k, k) binomial(2 n, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + k, k) binomial(2 n, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 8 7 6 5 4 b(n) = (517924 n - 4490860 n + 16142592 n - 31341035 n + 35864044 n 3 2 / 2 - 24744828 n + 10070689 n - 2220117 n + 204750) b(n - 1) / (n %1 / 2 7 6 5 4 (2 n - 1) ) + (2 n - 3) (347600 n - 2840200 n + 9364704 n - 16005762 n 3 2 / 2 + 15130773 n - 7808334 n + 2025433 n - 207194) b(n - 2) / (n %1 / 2 (2 n - 1) ) + (2 n - 5) (2 n - 3) 4 3 2 2 / 2 (3476 n - 9284 n + 8148 n - 2847 n + 351) (n - 2) b(n - 3) / (n %1 / 2 (2 n - 1) ) 4 3 2 %1 := 3476 n - 23188 n + 56856 n - 60899 n + 24106 and in Maple notation b(n) = (517924*n^8-4490860*n^7+16142592*n^6-31341035*n^5+35864044*n^4-24744828* n^3+10070689*n^2-2220117*n+204750)/n^2/(3476*n^4-23188*n^3+56856*n^2-60899*n+ 24106)/(2*n-1)^2*b(n-1)+(2*n-3)*(347600*n^7-2840200*n^6+9364704*n^5-16005762*n^ 4+15130773*n^3-7808334*n^2+2025433*n-207194)/n^2/(3476*n^4-23188*n^3+56856*n^2-\ 60899*n+24106)/(2*n-1)^2*b(n-2)+(2*n-5)*(2*n-3)*(3476*n^4-9284*n^3+8148*n^2-\ 2847*n+351)*(n-2)^2/n^2/(3476*n^4-23188*n^3+56856*n^2-60899*n+24106)/(2*n-1)^2* b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 9, b(2) = 193 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 8 7 6 5 4 a(n) = (517924 n - 4490860 n + 16142592 n - 31341035 n + 35864044 n 3 2 / 2 - 24744828 n + 10070689 n - 2220117 n + 204750) a(n - 1) / (n %1 / 2 7 6 5 4 (2 n - 1) ) + (2 n - 3) (347600 n - 2840200 n + 9364704 n - 16005762 n 3 2 / 2 + 15130773 n - 7808334 n + 2025433 n - 207194) a(n - 2) / (n %1 / 2 (2 n - 1) ) + (2 n - 5) (2 n - 3) 4 3 2 2 / 2 (3476 n - 9284 n + 8148 n - 2847 n + 351) (n - 2) a(n - 3) / (n %1 / 2 (2 n - 1) ) 4 3 2 %1 := 3476 n - 23188 n + 56856 n - 60899 n + 24106 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.10663802777845312452572801868587331816546155913234696217241230915282978799\ 53376485547482041532951827 This constant is identified as, 2/13 ln(2) The implied delta is, -0.3952433249552788118673358318986997876400892276664200\ 779885995597374822737888469552259700099378448401 Since this is negative, there is no Apery-style irrationality proof of, 2/13 ln(2), but still a very fast way to compute it to many digits ----------------------- This took, 1.798, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + k, k) binomial(2 n, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + k, k) binomial(2 n, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 8 7 6 5 b(n) = 1/4 (6147099 n - 53004231 n + 189377076 n - 365393148 n 4 3 2 + 415600821 n - 285202285 n + 115586012 n - 25419000 n + 2342808) / 2 2 7 6 b(n - 1) / (n %1 (2 n - 1) ) + (2 n - 3) (4768092 n - 38729502 n / 5 4 3 2 + 126611520 n - 213747585 n + 198457095 n - 99729287 n + 24898929 n / 2 2 - 2423278) b(n - 2) / (n %1 (2 n - 1) ) + 4 (2 n - 5) (2 n - 3) / 4 3 2 2 / 2 (7791 n - 20433 n + 17388 n - 5806 n + 676) (n - 2) b(n - 3) / (n %1 / 2 (2 n - 1) ) 4 3 2 %1 := 7791 n - 51597 n + 125433 n - 133045 n + 52094 and in Maple notation b(n) = 1/4*(6147099*n^8-53004231*n^7+189377076*n^6-365393148*n^5+415600821*n^4-\ 285202285*n^3+115586012*n^2-25419000*n+2342808)/n^2/(7791*n^4-51597*n^3+125433* n^2-133045*n+52094)/(2*n-1)^2*b(n-1)+(2*n-3)*(4768092*n^7-38729502*n^6+ 126611520*n^5-213747585*n^4+198457095*n^3-99729287*n^2+24898929*n-2423278)/n^2/ (7791*n^4-51597*n^3+125433*n^2-133045*n+52094)/(2*n-1)^2*b(n-2)+4*(2*n-5)*(2*n-\ 3)*(7791*n^4-20433*n^3+17388*n^2-5806*n+676)*(n-2)^2/n^2/(7791*n^4-51597*n^3+ 125433*n^2-133045*n+52094)/(2*n-1)^2*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 13, b(2) = 397 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 8 7 6 5 a(n) = 1/4 (6147099 n - 53004231 n + 189377076 n - 365393148 n 4 3 2 + 415600821 n - 285202285 n + 115586012 n - 25419000 n + 2342808) / 2 2 7 6 a(n - 1) / (n %1 (2 n - 1) ) + (2 n - 3) (4768092 n - 38729502 n / 5 4 3 2 + 126611520 n - 213747585 n + 198457095 n - 99729287 n + 24898929 n / 2 2 - 2423278) a(n - 2) / (n %1 (2 n - 1) ) + 4 (2 n - 5) (2 n - 3) / 4 3 2 2 / 2 (7791 n - 20433 n + 17388 n - 5806 n + 676) (n - 2) a(n - 3) / (n %1 / 2 (2 n - 1) ) 4 3 2 %1 := 7791 n - 51597 n + 125433 n - 133045 n + 52094 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.06866326804175685571220282730765785654046815986392184073341839585234339332\ 616306042960098467585825555 This constant is identified as, 1/16 ln(3) The implied delta is, -0.6171763499291324152728837009681205660643491270280632\ 126361858320181632861695894207271015107929434481 Since this is negative, there is no Apery-style irrationality proof of, 1/16 ln(3), but still a very fast way to compute it to many digits ----------------------- This took, 2.098, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ ) binomial(n, k) binomial(n + k, k) binomial(2 n + k, k) / ----- k By Shalosh B. Ekhad Let ----- \ b(n) = ) binomial(n, k) binomial(n + k, k) binomial(2 n + k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 7 6 5 4 3 b(n) = 1/4 (1550992 n - 14280832 n + 52658517 n - 99944475 n + 104650800 n 2 / 2 - 60062211 n + 17466825 n - 1987200) b(n - 1) / ((2 n - 1) %1 n ) + / 8 7 6 5 4 1/16 (1919978 n - 22478233 n + 111238909 n - 302384968 n + 490319339 n 3 2 / 2 - 480558727 n + 273743102 n - 81119980 n + 9411840) b(n - 2) / (n / (2 n - 1) (2 n - 3) %1) + 1/4 4 3 2 2 2 (6254 n - 23187 n + 28449 n - 12881 n + 1872) (n - 2) (2 n - 5) / 2 b(n - 3) / (n (2 n - 1) (2 n - 3) %1) / 4 3 2 %1 := 6254 n - 48203 n + 135534 n - 164356 n + 72643 and in Maple notation b(n) = 1/4*(1550992*n^7-14280832*n^6+52658517*n^5-99944475*n^4+104650800*n^3-\ 60062211*n^2+17466825*n-1987200)/(2*n-1)/(6254*n^4-48203*n^3+135534*n^2-164356* n+72643)/n^2*b(n-1)+1/16*(1919978*n^8-22478233*n^7+111238909*n^6-302384968*n^5+ 490319339*n^4-480558727*n^3+273743102*n^2-81119980*n+9411840)/n^2/(2*n-1)/(2*n-\ 3)/(6254*n^4-48203*n^3+135534*n^2-164356*n+72643)*b(n-2)+1/4*(6254*n^4-23187*n^ 3+28449*n^2-12881*n+1872)*(n-2)^2*(2*n-5)^2/n^2/(2*n-1)/(2*n-3)/(6254*n^4-48203 *n^3+135534*n^2-164356*n+72643)*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 7, b(2) = 121 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 7 6 5 4 3 a(n) = 1/4 (1550992 n - 14280832 n + 52658517 n - 99944475 n + 104650800 n 2 / 2 - 60062211 n + 17466825 n - 1987200) a(n - 1) / ((2 n - 1) %1 n ) + / 8 7 6 5 4 1/16 (1919978 n - 22478233 n + 111238909 n - 302384968 n + 490319339 n 3 2 / 2 - 480558727 n + 273743102 n - 81119980 n + 9411840) a(n - 2) / (n / (2 n - 1) (2 n - 3) %1) + 1/4 4 3 2 2 2 (6254 n - 23187 n + 28449 n - 12881 n + 1872) (n - 2) (2 n - 5) / 2 a(n - 3) / (n (2 n - 1) (2 n - 3) %1) / 4 3 2 %1 := 6254 n - 48203 n + 135534 n - 164356 n + 72643 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.14218403703793749936763735824783109088728207884312928289654974553710638399\ 37835314063309388710602436 This constant is identified as, 8/39 ln(2) The implied delta is, -0.1397644934796739777192589250530690451212418977148508\ 699852376546668571947837870642619350259297785470 Since this is negative, there is no Apery-style irrationality proof of, 8/39 ln(2), but still a very fast way to compute it to many digits ----------------------- This took, 1.904, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + k, k) binomial(2 n + k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + k, k) binomial(2 n + k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 7 6 5 4 b(n) = 1/9 (10995616 n - 102385408 n + 379899242 n - 722715416 n 3 2 / + 756851703 n - 434184126 n + 126255441 n - 14375880) b(n - 1) / ( / 2 8 7 6 (2 n - 1) %1 n ) + 1/9 (1878068 n - 22182754 n + 110463938 n 5 4 3 2 - 301261113 n + 488266368 n - 475854094 n + 267504185 n - 77411173 n / 2 + 8718840) b(n - 2) / (n (2 n - 1) (2 n - 3) %1) + 1/9 / 4 3 2 2 2 (10492 n - 39990 n + 49074 n - 21257 n + 2916) (n - 2) (2 n - 5) / 2 b(n - 3) / (n (2 n - 1) (2 n - 3) %1) / 4 3 2 %1 := 10492 n - 81958 n + 231996 n - 281343 n + 123729 and in Maple notation b(n) = 1/9*(10995616*n^7-102385408*n^6+379899242*n^5-722715416*n^4+756851703*n^ 3-434184126*n^2+126255441*n-14375880)/(2*n-1)/(10492*n^4-81958*n^3+231996*n^2-\ 281343*n+123729)/n^2*b(n-1)+1/9*(1878068*n^8-22182754*n^7+110463938*n^6-\ 301261113*n^5+488266368*n^4-475854094*n^3+267504185*n^2-77411173*n+8718840)/n^2 /(2*n-1)/(2*n-3)/(10492*n^4-81958*n^3+231996*n^2-281343*n+123729)*b(n-2)+1/9*( 10492*n^4-39990*n^3+49074*n^2-21257*n+2916)*(n-2)^2*(2*n-5)^2/n^2/(2*n-1)/(2*n-\ 3)/(10492*n^4-81958*n^3+231996*n^2-281343*n+123729)*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 13, b(2) = 421 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 7 6 5 4 a(n) = 1/9 (10995616 n - 102385408 n + 379899242 n - 722715416 n 3 2 / + 756851703 n - 434184126 n + 126255441 n - 14375880) a(n - 1) / ( / 2 8 7 6 (2 n - 1) %1 n ) + 1/9 (1878068 n - 22182754 n + 110463938 n 5 4 3 2 - 301261113 n + 488266368 n - 475854094 n + 267504185 n - 77411173 n / 2 + 8718840) a(n - 2) / (n (2 n - 1) (2 n - 3) %1) + 1/9 / 4 3 2 2 2 (10492 n - 39990 n + 49074 n - 21257 n + 2916) (n - 2) (2 n - 5) / 2 a(n - 3) / (n (2 n - 1) (2 n - 3) %1) / 4 3 2 %1 := 10492 n - 81958 n + 231996 n - 281343 n + 123729 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.07682496785207325132214985345640299429785081707710416375844481931151381139\ 453112129283730211275411813 18 18 This constant is identified as, -- ln(3) - -- ln(2) 95 95 The implied delta is, -0.1261062000996385863432395163014365287650243098493960\ 407059662857188203367434121078038391779419606337 Since this is negative, there is no Apery-style irrationality proof of, 18 18 -- ln(3) - -- ln(2), but still a very fast way to compute it to many digits 95 95 ----------------------- This took, 2.673, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + k, k) binomial(2 n + k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + k, k) binomial(2 n + k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 7 6 5 4 b(n) = 1/16 (31465980 n - 301703220 n + 1139319891 n - 2186144157 n 3 2 / + 2296428414 n - 1317951317 n + 383051609 n - 43591680) b(n - 1) / ( / 2 8 7 6 (2 n - 1) %1 n ) + 1/64 (13589154 n - 164268891 n + 832976811 n 5 4 3 2 - 2301585186 n + 3757274325 n - 3659327299 n + 2031351590 n / 2 - 569950684 n + 61332480) b(n - 2) / (n (2 n - 1) (2 n - 3) %1) + 1/16 / 4 3 2 2 2 (11526 n - 47121 n + 58659 n - 23669 n + 2880) (n - 2) (2 n - 5) / 2 b(n - 3) / (n (2 n - 1) (2 n - 3) %1) / 4 3 2 %1 := 11526 n - 93225 n + 269178 n - 328454 n + 143855 and in Maple notation b(n) = 1/16*(31465980*n^7-301703220*n^6+1139319891*n^5-2186144157*n^4+ 2296428414*n^3-1317951317*n^2+383051609*n-43591680)/(2*n-1)/(11526*n^4-93225*n^ 3+269178*n^2-328454*n+143855)/n^2*b(n-1)+1/64*(13589154*n^8-164268891*n^7+ 832976811*n^6-2301585186*n^5+3757274325*n^4-3659327299*n^3+2031351590*n^2-\ 569950684*n+61332480)/n^2/(2*n-1)/(2*n-3)/(11526*n^4-93225*n^3+269178*n^2-\ 328454*n+143855)*b(n-2)+1/16*(11526*n^4-47121*n^3+58659*n^2-23669*n+2880)*(n-2) ^2*(2*n-5)^2/n^2/(2*n-1)/(2*n-3)/(11526*n^4-93225*n^3+269178*n^2-328454*n+ 143855)*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 19, b(2) = 901 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 7 6 5 4 a(n) = 1/16 (31465980 n - 301703220 n + 1139319891 n - 2186144157 n 3 2 / + 2296428414 n - 1317951317 n + 383051609 n - 43591680) a(n - 1) / ( / 2 8 7 6 (2 n - 1) %1 n ) + 1/64 (13589154 n - 164268891 n + 832976811 n 5 4 3 2 - 2301585186 n + 3757274325 n - 3659327299 n + 2031351590 n / 2 - 569950684 n + 61332480) a(n - 2) / (n (2 n - 1) (2 n - 3) %1) + 1/16 / 4 3 2 2 2 (11526 n - 47121 n + 58659 n - 23669 n + 2880) (n - 2) (2 n - 5) / 2 a(n - 3) / (n (2 n - 1) (2 n - 3) %1) / 4 3 2 %1 := 11526 n - 93225 n + 269178 n - 328454 n + 143855 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.05260472181975422673174290395315701604635606142130487890407601103529928241\ 751414205039742156417925235 64 32 This constant is identified as, --- ln(2) - --- ln(3) 175 175 The implied delta is, -0.1233609432936533445508772225542104805516216781260622\ 662600559913689578199533266479555419346883068194 Since this is negative, there is no Apery-style irrationality proof of, 64 32 --- ln(2) - --- ln(3), 175 175 but still a very fast way to compute it to many digits ----------------------- This took, 2.708, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ ) binomial(n, k) binomial(n + k, k) binomial(2 n + 2 k, k) / ----- k By Shalosh B. Ekhad Let ----- \ b(n) = ) binomial(n, k) binomial(n + k, k) binomial(2 n + 2 k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 12 11 10 9 b(n) = 2 (177681600 n - 3583245600 n + 31921648560 n - 165636440820 n 8 7 6 5 + 555762364704 n - 1265811007212 n + 1998716464360 n - 2194729747951 n 4 3 2 + 1655149213612 n - 831588135547 n + 262715987704 n - 46557144850 n / 2 12 + 3469502400) b(n - 1) / (n (3 n - 1) (3 n - 2) %1) + 4 (124377120 n / 11 10 9 8 - 2632649040 n + 24747705360 n - 136246588524 n + 487685705184 n 7 6 5 - 1190923925328 n + 2024460625736 n - 2399051656581 n 4 3 2 + 1952209618374 n - 1054388543289 n + 355274766311 n - 66346853633 n / 2 12 + 5059115040) b(n - 2) / (n (3 n - 1) (3 n - 2) %1) + 4 (11845440 n / 11 10 9 8 - 262573920 n + 2590213680 n - 14985980808 n + 56407914172 n 7 6 5 4 - 144814599572 n + 258463659532 n - 320780483878 n + 272339218409 n 3 2 - 152658728731 n + 53043224652 n - 10148295046 n + 787224480) b(n - 3) / 2 8 7 / (n (3 n - 1) (3 n - 2) %1) + 4/3 (987120 n - 10035720 n / 6 5 4 3 2 + 42172860 n - 94880864 n + 123757560 n - 94721666 n + 41112927 n 2 2 / 2 - 9222968 n + 793440) (n - 3) (2 n - 7) b(n - 4) / (n (3 n - 1) / (3 n - 2) %1) 8 7 6 5 4 %1 := 987120 n - 17932680 n + 140062260 n - 613946864 n + 1651103380 n 3 2 - 2788546666 n + 2887614305 n - 1676232540 n + 417685125 and in Maple notation b(n) = 2*(177681600*n^12-3583245600*n^11+31921648560*n^10-165636440820*n^9+ 555762364704*n^8-1265811007212*n^7+1998716464360*n^6-2194729747951*n^5+ 1655149213612*n^4-831588135547*n^3+262715987704*n^2-46557144850*n+3469502400)/n ^2/(3*n-1)/(3*n-2)/(987120*n^8-17932680*n^7+140062260*n^6-613946864*n^5+ 1651103380*n^4-2788546666*n^3+2887614305*n^2-1676232540*n+417685125)*b(n-1)+4*( 124377120*n^12-2632649040*n^11+24747705360*n^10-136246588524*n^9+487685705184*n ^8-1190923925328*n^7+2024460625736*n^6-2399051656581*n^5+1952209618374*n^4-\ 1054388543289*n^3+355274766311*n^2-66346853633*n+5059115040)/n^2/(3*n-1)/(3*n-2 )/(987120*n^8-17932680*n^7+140062260*n^6-613946864*n^5+1651103380*n^4-\ 2788546666*n^3+2887614305*n^2-1676232540*n+417685125)*b(n-2)+4*(11845440*n^12-\ 262573920*n^11+2590213680*n^10-14985980808*n^9+56407914172*n^8-144814599572*n^7 +258463659532*n^6-320780483878*n^5+272339218409*n^4-152658728731*n^3+ 53043224652*n^2-10148295046*n+787224480)/n^2/(3*n-1)/(3*n-2)/(987120*n^8-\ 17932680*n^7+140062260*n^6-613946864*n^5+1651103380*n^4-2788546666*n^3+ 2887614305*n^2-1676232540*n+417685125)*b(n-3)+4/3*(987120*n^8-10035720*n^7+ 42172860*n^6-94880864*n^5+123757560*n^4-94721666*n^3+41112927*n^2-9222968*n+ 793440)*(n-3)^2*(2*n-7)^2/n^2/(3*n-1)/(3*n-2)/(987120*n^8-17932680*n^7+ 140062260*n^6-613946864*n^5+1651103380*n^4-2788546666*n^3+2887614305*n^2-\ 1676232540*n+417685125)*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 9, b(2) = 205, b(3) = 5847 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 12 11 10 9 a(n) = 2 (177681600 n - 3583245600 n + 31921648560 n - 165636440820 n 8 7 6 5 + 555762364704 n - 1265811007212 n + 1998716464360 n - 2194729747951 n 4 3 2 + 1655149213612 n - 831588135547 n + 262715987704 n - 46557144850 n / 2 12 + 3469502400) a(n - 1) / (n (3 n - 1) (3 n - 2) %1) + 4 (124377120 n / 11 10 9 8 - 2632649040 n + 24747705360 n - 136246588524 n + 487685705184 n 7 6 5 - 1190923925328 n + 2024460625736 n - 2399051656581 n 4 3 2 + 1952209618374 n - 1054388543289 n + 355274766311 n - 66346853633 n / 2 12 + 5059115040) a(n - 2) / (n (3 n - 1) (3 n - 2) %1) + 4 (11845440 n / 11 10 9 8 - 262573920 n + 2590213680 n - 14985980808 n + 56407914172 n 7 6 5 4 - 144814599572 n + 258463659532 n - 320780483878 n + 272339218409 n 3 2 - 152658728731 n + 53043224652 n - 10148295046 n + 787224480) a(n - 3) / 2 8 7 / (n (3 n - 1) (3 n - 2) %1) + 4/3 (987120 n - 10035720 n / 6 5 4 3 2 + 42172860 n - 94880864 n + 123757560 n - 94721666 n + 41112927 n 2 2 / 2 - 9222968 n + 793440) (n - 3) (2 n - 7) a(n - 4) / (n (3 n - 1) / (3 n - 2) %1) 8 7 6 5 4 %1 := 987120 n - 17932680 n + 140062260 n - 613946864 n + 1651103380 n 3 2 - 2788546666 n + 2887614305 n - 1676232540 n + 417685125 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.10797426194481728723446423468729508139620973944887864620421798218676457282\ 61422947212229336540424749 The implied delta is, -0.4807817961072793312972636352796199567996961791969705\ 981004426002271490936885155378610576244663400939 Since this is negative, there is no Apery-style irrationality proof of, 0.107\ 974261944817287234464234687295081396209739448878646204217982186764572826\ 1422947212229336540424749, but still a very fast way to compute it to many digits ----------------------- This took, 5.697, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + k, k) binomial(2 n + 2 k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + k, k) binomial(2 n + 2 k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 12 11 10 b(n) = 1/3 (5256534528 n - 106006779648 n + 944450613920 n 9 8 7 - 4901494569120 n + 16450777877768 n - 37483540619604 n 6 5 4 + 59217782106360 n - 65069148660780 n + 49113984018709 n 3 2 - 24703523077305 n + 7815776000647 n - 1387781673315 n + 103678364160) / 2 12 b(n - 1) / (n (3 n - 1) (3 n - 2) %1) + 1/3 (2786358528 n / 11 10 9 - 58977922176 n + 554549695872 n - 3054760880512 n 8 7 6 + 10944788795872 n - 26765459466344 n + 45590445792680 n 5 4 3 - 54172067322618 n + 44236374310141 n - 23997721947708 n 2 / 2 + 8131580861618 n - 1531465881123 n + 119346474240) b(n - 2) / (n / 12 11 (3 n - 1) (3 n - 2) %1) + 1/3 (207494784 n - 4599467712 n 10 9 8 7 + 45362916960 n - 262352650352 n + 987016302272 n - 2532577777216 n 6 5 4 + 4517967268012 n - 5605707846258 n + 4759620563999 n 3 2 - 2669833962297 n + 929292477592 n - 178624549554 n + 14115588480) / 2 8 7 b(n - 3) / (n (3 n - 1) (3 n - 2) %1) + 1/3 (2470176 n - 25113456 n / 6 5 4 3 2 + 105568016 n - 237701120 n + 310504244 n - 238195006 n + 103708811 n 2 2 / 2 - 23387568 n + 2060160) (n - 3) (2 n - 7) b(n - 4) / (n (3 n - 1) / (3 n - 2) %1) 8 7 6 5 4 %1 := 2470176 n - 44874864 n + 350527136 n - 1536821648 n + 4134413364 n 3 2 - 6985884318 n + 7238398237 n - 4204876480 n + 1048708557 and in Maple notation b(n) = 1/3*(5256534528*n^12-106006779648*n^11+944450613920*n^10-4901494569120*n ^9+16450777877768*n^8-37483540619604*n^7+59217782106360*n^6-65069148660780*n^5+ 49113984018709*n^4-24703523077305*n^3+7815776000647*n^2-1387781673315*n+ 103678364160)/n^2/(3*n-1)/(3*n-2)/(2470176*n^8-44874864*n^7+350527136*n^6-\ 1536821648*n^5+4134413364*n^4-6985884318*n^3+7238398237*n^2-4204876480*n+ 1048708557)*b(n-1)+1/3*(2786358528*n^12-58977922176*n^11+554549695872*n^10-\ 3054760880512*n^9+10944788795872*n^8-26765459466344*n^7+45590445792680*n^6-\ 54172067322618*n^5+44236374310141*n^4-23997721947708*n^3+8131580861618*n^2-\ 1531465881123*n+119346474240)/n^2/(3*n-1)/(3*n-2)/(2470176*n^8-44874864*n^7+ 350527136*n^6-1536821648*n^5+4134413364*n^4-6985884318*n^3+7238398237*n^2-\ 4204876480*n+1048708557)*b(n-2)+1/3*(207494784*n^12-4599467712*n^11+45362916960 *n^10-262352650352*n^9+987016302272*n^8-2532577777216*n^7+4517967268012*n^6-\ 5605707846258*n^5+4759620563999*n^4-2669833962297*n^3+929292477592*n^2-\ 178624549554*n+14115588480)/n^2/(3*n-1)/(3*n-2)/(2470176*n^8-44874864*n^7+ 350527136*n^6-1536821648*n^5+4134413364*n^4-6985884318*n^3+7238398237*n^2-\ 4204876480*n+1048708557)*b(n-3)+1/3*(2470176*n^8-25113456*n^7+105568016*n^6-\ 237701120*n^5+310504244*n^4-238195006*n^3+103708811*n^2-23387568*n+2060160)*(n-\ 3)^2*(2*n-7)^2/n^2/(3*n-1)/(3*n-2)/(2470176*n^8-44874864*n^7+350527136*n^6-\ 1536821648*n^5+4134413364*n^4-6985884318*n^3+7238398237*n^2-4204876480*n+ 1048708557)*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 17, b(2) = 745, b(3) = 40793 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 12 11 10 a(n) = 1/3 (5256534528 n - 106006779648 n + 944450613920 n 9 8 7 - 4901494569120 n + 16450777877768 n - 37483540619604 n 6 5 4 + 59217782106360 n - 65069148660780 n + 49113984018709 n 3 2 - 24703523077305 n + 7815776000647 n - 1387781673315 n + 103678364160) / 2 12 a(n - 1) / (n (3 n - 1) (3 n - 2) %1) + 1/3 (2786358528 n / 11 10 9 - 58977922176 n + 554549695872 n - 3054760880512 n 8 7 6 + 10944788795872 n - 26765459466344 n + 45590445792680 n 5 4 3 - 54172067322618 n + 44236374310141 n - 23997721947708 n 2 / 2 + 8131580861618 n - 1531465881123 n + 119346474240) a(n - 2) / (n / 12 11 (3 n - 1) (3 n - 2) %1) + 1/3 (207494784 n - 4599467712 n 10 9 8 7 + 45362916960 n - 262352650352 n + 987016302272 n - 2532577777216 n 6 5 4 + 4517967268012 n - 5605707846258 n + 4759620563999 n 3 2 - 2669833962297 n + 929292477592 n - 178624549554 n + 14115588480) / 2 8 7 a(n - 3) / (n (3 n - 1) (3 n - 2) %1) + 1/3 (2470176 n - 25113456 n / 6 5 4 3 2 + 105568016 n - 237701120 n + 310504244 n - 238195006 n + 103708811 n 2 2 / 2 - 23387568 n + 2060160) (n - 3) (2 n - 7) a(n - 4) / (n (3 n - 1) / (3 n - 2) %1) 8 7 6 5 4 %1 := 2470176 n - 44874864 n + 350527136 n - 1536821648 n + 4134413364 n 3 2 - 6985884318 n + 7238398237 n - 4204876480 n + 1048708557 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.05846355282424832904966496043275291207821066409728334671985768900850474877\ 348392502316928006762184608 7232 This constant is identified as, ----- ln(2) 85743 The implied delta is, -0.4085139225203998850639468721750085492974607039826882\ 081229693556362255488006979246998318769922634482 Since this is negative, there is no Apery-style irrationality proof of, 7232 ----- ln(2), but still a very fast way to compute it to many digits 85743 ----------------------- This took, 5.476, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + k, k) binomial(2 n + 2 k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + k, k) binomial(2 n + 2 k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 12 11 10 b(n) = 2/27 (349508098368 n - 7048413317088 n + 62798668380144 n 9 8 7 - 325933011591988 n + 1094035605896056 n - 2493150724091288 n 6 5 4 + 3939510428240836 n - 4329820091652365 n + 3269137797047608 n 3 2 - 1644978086820253 n + 520720850045568 n - 92528674644078 n / 2 + 6920097468480) b(n - 1) / (n (3 n - 1) (3 n - 2) %1) + 4/9 ( / 12 11 10 9 18542717856 n - 392487527952 n + 3690820529760 n - 20335892366284 n 8 7 6 + 72889922949208 n - 178358536876984 n + 304055121811564 n 5 4 3 - 361682271242109 n + 295750920826450 n - 160703018316209 n 2 / 2 + 54553118025003 n - 10296221473613 n + 806253950880) b(n - 2) / (n / 12 11 (3 n - 1) (3 n - 2) %1) + 4/9 (1082068416 n - 23985849888 n 10 9 8 + 236551442448 n - 1367947091096 n + 5145844261092 n 7 6 5 - 13202081372924 n + 23549461379736 n - 29218075280302 n 4 3 2 + 24809315235249 n - 13918519524463 n + 4845932283332 n / 2 - 932076918430 n + 73932717600) b(n - 3) / (n (3 n - 1) (3 n - 2) %1) / 8 7 6 5 + 4/27 (24592464 n - 250023384 n + 1051124796 n - 2367390424 n 4 3 2 + 3093926160 n - 2374979794 n + 1034777043 n - 233524264 n + 20671200) 2 2 / 2 (n - 3) (2 n - 7) b(n - 4) / (n (3 n - 1) (3 n - 2) %1) / 8 7 6 5 %1 := 24592464 n - 446763096 n + 3489877476 n - 15301808248 n 4 3 2 + 41170041140 n - 69575081018 n + 72103129621 n - 41894326668 n + 10451009529 and in Maple notation b(n) = 2/27*(349508098368*n^12-7048413317088*n^11+62798668380144*n^10-\ 325933011591988*n^9+1094035605896056*n^8-2493150724091288*n^7+3939510428240836* n^6-4329820091652365*n^5+3269137797047608*n^4-1644978086820253*n^3+ 520720850045568*n^2-92528674644078*n+6920097468480)/n^2/(3*n-1)/(3*n-2)/( 24592464*n^8-446763096*n^7+3489877476*n^6-15301808248*n^5+41170041140*n^4-\ 69575081018*n^3+72103129621*n^2-41894326668*n+10451009529)*b(n-1)+4/9*( 18542717856*n^12-392487527952*n^11+3690820529760*n^10-20335892366284*n^9+ 72889922949208*n^8-178358536876984*n^7+304055121811564*n^6-361682271242109*n^5+ 295750920826450*n^4-160703018316209*n^3+54553118025003*n^2-10296221473613*n+ 806253950880)/n^2/(3*n-1)/(3*n-2)/(24592464*n^8-446763096*n^7+3489877476*n^6-\ 15301808248*n^5+41170041140*n^4-69575081018*n^3+72103129621*n^2-41894326668*n+ 10451009529)*b(n-2)+4/9*(1082068416*n^12-23985849888*n^11+236551442448*n^10-\ 1367947091096*n^9+5145844261092*n^8-13202081372924*n^7+23549461379736*n^6-\ 29218075280302*n^5+24809315235249*n^4-13918519524463*n^3+4845932283332*n^2-\ 932076918430*n+73932717600)/n^2/(3*n-1)/(3*n-2)/(24592464*n^8-446763096*n^7+ 3489877476*n^6-15301808248*n^5+41170041140*n^4-69575081018*n^3+72103129621*n^2-\ 41894326668*n+10451009529)*b(n-3)+4/27*(24592464*n^8-250023384*n^7+1051124796*n ^6-2367390424*n^5+3093926160*n^4-2374979794*n^3+1034777043*n^2-233524264*n+ 20671200)*(n-3)^2*(2*n-7)^2/n^2/(3*n-1)/(3*n-2)/(24592464*n^8-446763096*n^7+ 3489877476*n^6-15301808248*n^5+41170041140*n^4-69575081018*n^3+72103129621*n^2-\ 41894326668*n+10451009529)*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 25, b(2) = 1621, b(3) = 131239 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 12 11 10 a(n) = 2/27 (349508098368 n - 7048413317088 n + 62798668380144 n 9 8 7 - 325933011591988 n + 1094035605896056 n - 2493150724091288 n 6 5 4 + 3939510428240836 n - 4329820091652365 n + 3269137797047608 n 3 2 - 1644978086820253 n + 520720850045568 n - 92528674644078 n / 2 + 6920097468480) a(n - 1) / (n (3 n - 1) (3 n - 2) %1) + 4/9 ( / 12 11 10 9 18542717856 n - 392487527952 n + 3690820529760 n - 20335892366284 n 8 7 6 + 72889922949208 n - 178358536876984 n + 304055121811564 n 5 4 3 - 361682271242109 n + 295750920826450 n - 160703018316209 n 2 / 2 + 54553118025003 n - 10296221473613 n + 806253950880) a(n - 2) / (n / 12 11 (3 n - 1) (3 n - 2) %1) + 4/9 (1082068416 n - 23985849888 n 10 9 8 + 236551442448 n - 1367947091096 n + 5145844261092 n 7 6 5 - 13202081372924 n + 23549461379736 n - 29218075280302 n 4 3 2 + 24809315235249 n - 13918519524463 n + 4845932283332 n / 2 - 932076918430 n + 73932717600) a(n - 3) / (n (3 n - 1) (3 n - 2) %1) / 8 7 6 5 + 4/27 (24592464 n - 250023384 n + 1051124796 n - 2367390424 n 4 3 2 + 3093926160 n - 2374979794 n + 1034777043 n - 233524264 n + 20671200) 2 2 / 2 (n - 3) (2 n - 7) a(n - 4) / (n (3 n - 1) (3 n - 2) %1) / 8 7 6 5 %1 := 24592464 n - 446763096 n + 3489877476 n - 15301808248 n 4 3 2 + 41170041140 n - 69575081018 n + 72103129621 n - 41894326668 n + 10451009529 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.03989846749407357018501950276168786773599726272300818869750196531616513157\ 617454101909415482527895688 The implied delta is, -0.3829191213060336346358215568378287000110470872875984\ 733572223157281457396784769951472749240429191337 Since this is negative, there is no Apery-style irrationality proof of, 0.039\ 898467494073570185019502761687867735997262723008188697501965316165131576\ 17454101909415482527895688, but still a very fast way to compute it to many digits ----------------------- This took, 5.196, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n, k) / ----- k By Shalosh B. Ekhad Let ----- \ b(n) = ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 16 15 14 b(n) = 1/12 (71491338804510 n - 1666398758256855 n + 17733671479303476 n 13 12 - 114242658666838228 n + 498024303580605992 n 11 10 - 1555654188021881918 n + 3596380511620656814 n 9 8 7 - 6266810657287698390 n + 8304801922105780422 n - 8383777746638443415 n 6 5 4 + 6415749765418823194 n - 3677530526675182354 n + 1546235775842192984 n 3 2 - 460761985033790904 n + 91798712800380864 n - 10941481772141760 n / 2 2 + 589606825248000) b(n - 1) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) + / 16 15 14 1/2 (33327126658692 n - 810152451362358 n + 9021808000975896 n 13 12 11 - 61023483364051689 n + 280237658895134256 n - 925004215217154042 n 10 9 + 2265884239085170355 n - 4192604375838509701 n 8 7 6 + 5906854625375559772 n - 6338784287323213530 n + 5146515481850746539 n 5 4 3 - 3116428663686158664 n + 1374409197919819474 n - 425277514480852544 n 2 + 86901970481679464 n - 10494359989960320 n + 568738523404800) b(n - 2) / 2 2 13 / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) - (2 n - 5) (201778768872 n / 12 11 10 - 3795277021632 n + 31677290925939 n - 154962033789942 n 9 8 7 + 494373298749519 n - 1082979961655036 n + 1670430486180844 n 6 5 4 - 1830061650137146 n + 1418223180977743 n - 764557024906372 n 3 2 + 277672357526707 n - 64251451737416 n + 8526011634072 n - 496946677752) 2 / 2 2 (n - 2) b(n - 3) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) + 9 (2 n - 5) / 10 9 8 (2 n - 7) (5604965802 n - 57782152395 n + 256095796368 n 7 6 5 4 - 641154450536 n + 1000788342896 n - 1013512384085 n + 671477047172 n 3 2 2 - 286402898264 n + 75249101426 n - 11066945328 n + 703318824) (n - 2) 2 / 2 2 (n - 3) b(n - 4) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) / 10 9 8 7 %1 := 5604965802 n - 113831810415 n + 1028358629013 n - 5442674203940 n 6 5 4 + 18690295414552 n - 43516853083199 n + 69575569645747 n 3 2 - 75431268848510 n + 53077575770410 n - 21891910563732 n + 4019837403096 and in Maple notation b(n) = 1/12*(71491338804510*n^16-1666398758256855*n^15+17733671479303476*n^14-\ 114242658666838228*n^13+498024303580605992*n^12-1555654188021881918*n^11+ 3596380511620656814*n^10-6266810657287698390*n^9+8304801922105780422*n^8-\ 8383777746638443415*n^7+6415749765418823194*n^6-3677530526675182354*n^5+ 1546235775842192984*n^4-460761985033790904*n^3+91798712800380864*n^2-\ 10941481772141760*n+589606825248000)/n^2/(3*n-1)/(3*n-2)/(5604965802*n^10-\ 113831810415*n^9+1028358629013*n^8-5442674203940*n^7+18690295414552*n^6-\ 43516853083199*n^5+69575569645747*n^4-75431268848510*n^3+53077575770410*n^2-\ 21891910563732*n+4019837403096)/(2*n-1)^2*b(n-1)+1/2*(33327126658692*n^16-\ 810152451362358*n^15+9021808000975896*n^14-61023483364051689*n^13+ 280237658895134256*n^12-925004215217154042*n^11+2265884239085170355*n^10-\ 4192604375838509701*n^9+5906854625375559772*n^8-6338784287323213530*n^7+ 5146515481850746539*n^6-3116428663686158664*n^5+1374409197919819474*n^4-\ 425277514480852544*n^3+86901970481679464*n^2-10494359989960320*n+ 568738523404800)/n^2/(3*n-1)/(3*n-2)/(5604965802*n^10-113831810415*n^9+ 1028358629013*n^8-5442674203940*n^7+18690295414552*n^6-43516853083199*n^5+ 69575569645747*n^4-75431268848510*n^3+53077575770410*n^2-21891910563732*n+ 4019837403096)/(2*n-1)^2*b(n-2)-(2*n-5)*(201778768872*n^13-3795277021632*n^12+ 31677290925939*n^11-154962033789942*n^10+494373298749519*n^9-1082979961655036*n ^8+1670430486180844*n^7-1830061650137146*n^6+1418223180977743*n^5-\ 764557024906372*n^4+277672357526707*n^3-64251451737416*n^2+8526011634072*n-\ 496946677752)*(n-2)^2/n^2/(3*n-1)/(3*n-2)/(5604965802*n^10-113831810415*n^9+ 1028358629013*n^8-5442674203940*n^7+18690295414552*n^6-43516853083199*n^5+ 69575569645747*n^4-75431268848510*n^3+53077575770410*n^2-21891910563732*n+ 4019837403096)/(2*n-1)^2*b(n-3)+9*(2*n-5)*(2*n-7)*(5604965802*n^10-57782152395* n^9+256095796368*n^8-641154450536*n^7+1000788342896*n^6-1013512384085*n^5+ 671477047172*n^4-286402898264*n^3+75249101426*n^2-11066945328*n+703318824)*(n-2 )^2*(n-3)^2/n^2/(3*n-1)/(3*n-2)/(5604965802*n^10-113831810415*n^9+1028358629013 *n^8-5442674203940*n^7+18690295414552*n^6-43516853083199*n^5+69575569645747*n^4 -75431268848510*n^3+53077575770410*n^2-21891910563732*n+4019837403096)/(2*n-1)^ 2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 7, b(2) = 123, b(3) = 2716 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 16 15 14 a(n) = 1/12 (71491338804510 n - 1666398758256855 n + 17733671479303476 n 13 12 - 114242658666838228 n + 498024303580605992 n 11 10 - 1555654188021881918 n + 3596380511620656814 n 9 8 7 - 6266810657287698390 n + 8304801922105780422 n - 8383777746638443415 n 6 5 4 + 6415749765418823194 n - 3677530526675182354 n + 1546235775842192984 n 3 2 - 460761985033790904 n + 91798712800380864 n - 10941481772141760 n / 2 2 + 589606825248000) a(n - 1) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) + / 16 15 14 1/2 (33327126658692 n - 810152451362358 n + 9021808000975896 n 13 12 11 - 61023483364051689 n + 280237658895134256 n - 925004215217154042 n 10 9 + 2265884239085170355 n - 4192604375838509701 n 8 7 6 + 5906854625375559772 n - 6338784287323213530 n + 5146515481850746539 n 5 4 3 - 3116428663686158664 n + 1374409197919819474 n - 425277514480852544 n 2 + 86901970481679464 n - 10494359989960320 n + 568738523404800) a(n - 2) / 2 2 13 / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) - (2 n - 5) (201778768872 n / 12 11 10 - 3795277021632 n + 31677290925939 n - 154962033789942 n 9 8 7 + 494373298749519 n - 1082979961655036 n + 1670430486180844 n 6 5 4 - 1830061650137146 n + 1418223180977743 n - 764557024906372 n 3 2 + 277672357526707 n - 64251451737416 n + 8526011634072 n - 496946677752) 2 / 2 2 (n - 2) a(n - 3) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) + 9 (2 n - 5) / 10 9 8 (2 n - 7) (5604965802 n - 57782152395 n + 256095796368 n 7 6 5 4 - 641154450536 n + 1000788342896 n - 1013512384085 n + 671477047172 n 3 2 2 - 286402898264 n + 75249101426 n - 11066945328 n + 703318824) (n - 2) 2 / 2 2 (n - 3) a(n - 4) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) / 10 9 8 7 %1 := 5604965802 n - 113831810415 n + 1028358629013 n - 5442674203940 n 6 5 4 + 18690295414552 n - 43516853083199 n + 69575569645747 n 3 2 - 75431268848510 n + 53077575770410 n - 21891910563732 n + 4019837403096 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately -0.1211691924130834951236694740442687531745740599126002343437847448859723297\ -2182 331172716026755910862306067 10 This constant is identified as, 0 The implied delta is, -0.5387894636305083296223914558830495625807614351949514\ 140637866548377784069234017930212018579533038314 Since this is negative, there is no Apery-style irrationality proof of, 0, but still a very fast way to compute it to many digits ----------------------- This took, 1.754, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 16 15 b(n) = 1/2 (1785513894476976 n - 41433592692573504 n 14 13 + 438853913247935232 n - 2813212811276760060 n 12 11 + 12201583588188398472 n - 37919090114350779980 n 10 9 + 87226649144719531488 n - 151296842342675336521 n 8 7 + 199716162597592307147 n - 201053624828789498518 n 6 5 + 153685756453599389450 n - 88199503362453504089 n 4 3 + 37240707837520654295 n - 11184080281473223008 n 2 + 2253886009712089980 n - 272495913427827360 n + 14900989869360000) / 2 2 b(n - 1) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) + 1/2 ( / 16 15 14 45017402710750344 n - 1089664959789078120 n + 12074958450717266628 n 13 12 - 81213230646565825266 n + 370512406796433183000 n 11 10 - 1213654960626755491613 n + 2946396878539259989885 n 9 8 - 5394293959327123456631 n + 7504611647054725611990 n 7 6 - 7932288774018612968214 n + 6323094981846071270322 n 5 4 - 3743796119859961014328 n + 1605933556921061441051 n 3 2 - 480145808477449353028 n + 94064468890789378620 n / 2 - 10806732717402776640 n + 555276148770672000) b(n - 2) / (n (3 n - 1) / 2 13 (3 n - 2) %1 (2 n - 1) ) - 1/2 (2 n - 5) (313623199029888 n 12 11 10 - 5866452404657856 n + 48655481661979632 n - 236278972563077772 n 9 8 7 + 747325617782646426 n - 1620320635569441514 n + 2468231539538925779 n 6 5 4 - 2662996907348923915 n + 2024932720705704167 n - 1066111461679150489 n 3 2 + 375911775607848656 n - 83856693919385574 n + 10654364566288812 n 2 / 2 - 593422099626240) (n - 2) b(n - 3) / (n (3 n - 1) (3 n - 2) %1 / 2 10 (2 n - 1) ) + 343/6 (2 n - 5) (2 n - 7) (593983331496 n 9 8 7 - 6061846994136 n + 26530768127364 n - 65389971454282 n 6 5 4 + 100090218303658 n - 98896920866911 n + 63509646711946 n 3 2 - 26037091592371 n + 6509042191456 n - 901933857420 n + 53869320000) 2 2 / 2 2 (n - 2) (n - 3) b(n - 4) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) / 10 9 8 %1 := 593983331496 n - 12001680309096 n + 107816640991908 n 7 6 5 - 567140608041610 n + 1934613173171408 n - 4471827167159293 n 4 3 2 + 7093679516292017 n - 7625694416501623 n + 5317007330276555 n - 2171568195482802 n + 394575292751040 and in Maple notation b(n) = 1/2*(1785513894476976*n^16-41433592692573504*n^15+438853913247935232*n^ 14-2813212811276760060*n^13+12201583588188398472*n^12-37919090114350779980*n^11 +87226649144719531488*n^10-151296842342675336521*n^9+199716162597592307147*n^8-\ 201053624828789498518*n^7+153685756453599389450*n^6-88199503362453504089*n^5+ 37240707837520654295*n^4-11184080281473223008*n^3+2253886009712089980*n^2-\ 272495913427827360*n+14900989869360000)/n^2/(3*n-1)/(3*n-2)/(593983331496*n^10-\ 12001680309096*n^9+107816640991908*n^8-567140608041610*n^7+1934613173171408*n^6 -4471827167159293*n^5+7093679516292017*n^4-7625694416501623*n^3+ 5317007330276555*n^2-2171568195482802*n+394575292751040)/(2*n-1)^2*b(n-1)+1/2*( 45017402710750344*n^16-1089664959789078120*n^15+12074958450717266628*n^14-\ 81213230646565825266*n^13+370512406796433183000*n^12-1213654960626755491613*n^ 11+2946396878539259989885*n^10-5394293959327123456631*n^9+ 7504611647054725611990*n^8-7932288774018612968214*n^7+6323094981846071270322*n^ 6-3743796119859961014328*n^5+1605933556921061441051*n^4-480145808477449353028*n ^3+94064468890789378620*n^2-10806732717402776640*n+555276148770672000)/n^2/(3*n -1)/(3*n-2)/(593983331496*n^10-12001680309096*n^9+107816640991908*n^8-\ 567140608041610*n^7+1934613173171408*n^6-4471827167159293*n^5+7093679516292017* n^4-7625694416501623*n^3+5317007330276555*n^2-2171568195482802*n+ 394575292751040)/(2*n-1)^2*b(n-2)-1/2*(2*n-5)*(313623199029888*n^13-\ 5866452404657856*n^12+48655481661979632*n^11-236278972563077772*n^10+ 747325617782646426*n^9-1620320635569441514*n^8+2468231539538925779*n^7-\ 2662996907348923915*n^6+2024932720705704167*n^5-1066111461679150489*n^4+ 375911775607848656*n^3-83856693919385574*n^2+10654364566288812*n-\ 593422099626240)*(n-2)^2/n^2/(3*n-1)/(3*n-2)/(593983331496*n^10-12001680309096* n^9+107816640991908*n^8-567140608041610*n^7+1934613173171408*n^6-\ 4471827167159293*n^5+7093679516292017*n^4-7625694416501623*n^3+5317007330276555 *n^2-2171568195482802*n+394575292751040)/(2*n-1)^2*b(n-3)+343/6*(2*n-5)*(2*n-7) *(593983331496*n^10-6061846994136*n^9+26530768127364*n^8-65389971454282*n^7+ 100090218303658*n^6-98896920866911*n^5+63509646711946*n^4-26037091592371*n^3+ 6509042191456*n^2-901933857420*n+53869320000)*(n-2)^2*(n-3)^2/n^2/(3*n-1)/(3*n-\ 2)/(593983331496*n^10-12001680309096*n^9+107816640991908*n^8-567140608041610*n^ 7+1934613173171408*n^6-4471827167159293*n^5+7093679516292017*n^4-\ 7625694416501623*n^3+5317007330276555*n^2-2171568195482802*n+394575292751040)/( 2*n-1)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 13, b(2) = 425, b(3) = 17401 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 16 15 a(n) = 1/2 (1785513894476976 n - 41433592692573504 n 14 13 + 438853913247935232 n - 2813212811276760060 n 12 11 + 12201583588188398472 n - 37919090114350779980 n 10 9 + 87226649144719531488 n - 151296842342675336521 n 8 7 + 199716162597592307147 n - 201053624828789498518 n 6 5 + 153685756453599389450 n - 88199503362453504089 n 4 3 + 37240707837520654295 n - 11184080281473223008 n 2 + 2253886009712089980 n - 272495913427827360 n + 14900989869360000) / 2 2 a(n - 1) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) + 1/2 ( / 16 15 14 45017402710750344 n - 1089664959789078120 n + 12074958450717266628 n 13 12 - 81213230646565825266 n + 370512406796433183000 n 11 10 - 1213654960626755491613 n + 2946396878539259989885 n 9 8 - 5394293959327123456631 n + 7504611647054725611990 n 7 6 - 7932288774018612968214 n + 6323094981846071270322 n 5 4 - 3743796119859961014328 n + 1605933556921061441051 n 3 2 - 480145808477449353028 n + 94064468890789378620 n / 2 - 10806732717402776640 n + 555276148770672000) a(n - 2) / (n (3 n - 1) / 2 13 (3 n - 2) %1 (2 n - 1) ) - 1/2 (2 n - 5) (313623199029888 n 12 11 10 - 5866452404657856 n + 48655481661979632 n - 236278972563077772 n 9 8 7 + 747325617782646426 n - 1620320635569441514 n + 2468231539538925779 n 6 5 4 - 2662996907348923915 n + 2024932720705704167 n - 1066111461679150489 n 3 2 + 375911775607848656 n - 83856693919385574 n + 10654364566288812 n 2 / 2 - 593422099626240) (n - 2) a(n - 3) / (n (3 n - 1) (3 n - 2) %1 / 2 10 (2 n - 1) ) + 343/6 (2 n - 5) (2 n - 7) (593983331496 n 9 8 7 - 6061846994136 n + 26530768127364 n - 65389971454282 n 6 5 4 + 100090218303658 n - 98896920866911 n + 63509646711946 n 3 2 - 26037091592371 n + 6509042191456 n - 901933857420 n + 53869320000) 2 2 / 2 2 (n - 2) (n - 3) a(n - 4) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) / 10 9 8 %1 := 593983331496 n - 12001680309096 n + 107816640991908 n 7 6 5 - 567140608041610 n + 1934613173171408 n - 4471827167159293 n 4 3 2 + 7093679516292017 n - 7625694416501623 n + 5317007330276555 n - 2171568195482802 n + 394575292751040 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.04780325383172036616670566354883976331555173340415553476694344893057887048\ 066860107626643634458059914 This constant is identified as, 2/29 ln(2) The implied delta is, -0.8001153806680037311784063279100046752675405310312556\ 781601345108896393629659834815999677067915159885 Since this is negative, there is no Apery-style irrationality proof of, 2/29 ln(2), but still a very fast way to compute it to many digits ----------------------- This took, 1.923, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 16 15 b(n) = 1/36 (3440596013277142122 n - 79641581326700220945 n 14 13 + 841490594252089447656 n - 5382241569653371502156 n 12 11 + 23301921238000471285828 n - 72338267837900201080318 n 10 9 + 166420250389128160999022 n - 289208405919656404502190 n 8 7 + 383485635135757388124102 n - 389208749369364366756565 n 6 5 + 301404002400081117144782 n - 176314029768923137582730 n 4 3 + 76426303510798336181512 n - 23737145158551717405048 n 2 + 4977645770060146862784 n - 627912666421118159040 n / 2 2 + 35710802707451270400) b(n - 1) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) / 16 15 ) + 1/6 (59175268776845093808 n - 1428941426336137989288 n 14 13 + 15790933048519512018582 n - 105866113109672709313441 n 12 11 + 481180023263983958525314 n - 1569236921816570162470244 n 10 9 + 3789808700108082813545675 n - 6895168573226787536558429 n 8 7 + 9520243716293846898714826 n - 9969624574405102717634810 n 6 5 + 7855539650756060370465593 n - 4583327399677915111789804 n 4 3 + 1929200051982706595339746 n - 562695352819937182980112 n 2 + 106707103209138208662344 n - 11756703635049646517760 n / 2 + 575125383831656140800) b(n - 2) / (n (3 n - 1) (3 n - 2) %1 / 2 13 (2 n - 1) ) - 1/3 (2 n - 5) (139324758020516484 n 12 11 - 2598074019947687112 n + 21467508676826505741 n 10 9 - 103775491836805587738 n + 326398842448508123199 n 8 7 - 702791655839154599876 n + 1061292148279194220834 n 6 5 - 1132460824865511151318 n + 848937553019285355871 n 4 3 - 438678803033019022468 n + 150867112811963954839 n 2 - 32545165522154817560 n + 3957146162230448808 n - 209303570148739704) 2 / 2 2 (n - 2) b(n - 3) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) + 1331/9 / 10 9 (2 n - 5) (2 n - 7) (67176836075466 n - 681684463246527 n 8 7 6 + 2962236507130380 n - 7235043050875940 n + 10946358262282208 n 5 4 3 - 10653200765574773 n + 6705118693140104 n - 2675288529923864 n 2 2 2 + 644547097664306 n - 85054613015232 n + 4798977647112) (n - 2) (n - 3) / 2 2 b(n - 4) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) / 10 9 8 %1 := 67176836075466 n - 1353452824001187 n + 12120354299745093 n 7 6 5 - 63534796113849872 n + 215902912306620556 n - 496973303867043875 n 4 3 2 + 784748936679941851 n - 839389402625599538 n + 582070631004573694 n - 236314764515390988 n + 42660507796575912 and in Maple notation b(n) = 1/36*(3440596013277142122*n^16-79641581326700220945*n^15+ 841490594252089447656*n^14-5382241569653371502156*n^13+23301921238000471285828* n^12-72338267837900201080318*n^11+166420250389128160999022*n^10-\ 289208405919656404502190*n^9+383485635135757388124102*n^8-\ 389208749369364366756565*n^7+301404002400081117144782*n^6-\ 176314029768923137582730*n^5+76426303510798336181512*n^4-\ 23737145158551717405048*n^3+4977645770060146862784*n^2-627912666421118159040*n+ 35710802707451270400)/n^2/(3*n-1)/(3*n-2)/(67176836075466*n^10-1353452824001187 *n^9+12120354299745093*n^8-63534796113849872*n^7+215902912306620556*n^6-\ 496973303867043875*n^5+784748936679941851*n^4-839389402625599538*n^3+ 582070631004573694*n^2-236314764515390988*n+42660507796575912)/(2*n-1)^2*b(n-1) +1/6*(59175268776845093808*n^16-1428941426336137989288*n^15+ 15790933048519512018582*n^14-105866113109672709313441*n^13+ 481180023263983958525314*n^12-1569236921816570162470244*n^11+ 3789808700108082813545675*n^10-6895168573226787536558429*n^9+ 9520243716293846898714826*n^8-9969624574405102717634810*n^7+ 7855539650756060370465593*n^6-4583327399677915111789804*n^5+ 1929200051982706595339746*n^4-562695352819937182980112*n^3+ 106707103209138208662344*n^2-11756703635049646517760*n+575125383831656140800)/n ^2/(3*n-1)/(3*n-2)/(67176836075466*n^10-1353452824001187*n^9+12120354299745093* n^8-63534796113849872*n^7+215902912306620556*n^6-496973303867043875*n^5+ 784748936679941851*n^4-839389402625599538*n^3+582070631004573694*n^2-\ 236314764515390988*n+42660507796575912)/(2*n-1)^2*b(n-2)-1/3*(2*n-5)*( 139324758020516484*n^13-2598074019947687112*n^12+21467508676826505741*n^11-\ 103775491836805587738*n^10+326398842448508123199*n^9-702791655839154599876*n^8+ 1061292148279194220834*n^7-1132460824865511151318*n^6+848937553019285355871*n^5 -438678803033019022468*n^4+150867112811963954839*n^3-32545165522154817560*n^2+ 3957146162230448808*n-209303570148739704)*(n-2)^2/n^2/(3*n-1)/(3*n-2)/( 67176836075466*n^10-1353452824001187*n^9+12120354299745093*n^8-\ 63534796113849872*n^7+215902912306620556*n^6-496973303867043875*n^5+ 784748936679941851*n^4-839389402625599538*n^3+582070631004573694*n^2-\ 236314764515390988*n+42660507796575912)/(2*n-1)^2*b(n-3)+1331/9*(2*n-5)*(2*n-7) *(67176836075466*n^10-681684463246527*n^9+2962236507130380*n^8-7235043050875940 *n^7+10946358262282208*n^6-10653200765574773*n^5+6705118693140104*n^4-\ 2675288529923864*n^3+644547097664306*n^2-85054613015232*n+4798977647112)*(n-2)^ 2*(n-3)^2/n^2/(3*n-1)/(3*n-2)/(67176836075466*n^10-1353452824001187*n^9+ 12120354299745093*n^8-63534796113849872*n^7+215902912306620556*n^6-\ 496973303867043875*n^5+784748936679941851*n^4-839389402625599538*n^3+ 582070631004573694*n^2-236314764515390988*n+42660507796575912)/(2*n-1)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 19, b(2) = 907, b(3) = 54136 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 16 15 a(n) = 1/36 (3440596013277142122 n - 79641581326700220945 n 14 13 + 841490594252089447656 n - 5382241569653371502156 n 12 11 + 23301921238000471285828 n - 72338267837900201080318 n 10 9 + 166420250389128160999022 n - 289208405919656404502190 n 8 7 + 383485635135757388124102 n - 389208749369364366756565 n 6 5 + 301404002400081117144782 n - 176314029768923137582730 n 4 3 + 76426303510798336181512 n - 23737145158551717405048 n 2 + 4977645770060146862784 n - 627912666421118159040 n / 2 2 + 35710802707451270400) a(n - 1) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) / 16 15 ) + 1/6 (59175268776845093808 n - 1428941426336137989288 n 14 13 + 15790933048519512018582 n - 105866113109672709313441 n 12 11 + 481180023263983958525314 n - 1569236921816570162470244 n 10 9 + 3789808700108082813545675 n - 6895168573226787536558429 n 8 7 + 9520243716293846898714826 n - 9969624574405102717634810 n 6 5 + 7855539650756060370465593 n - 4583327399677915111789804 n 4 3 + 1929200051982706595339746 n - 562695352819937182980112 n 2 + 106707103209138208662344 n - 11756703635049646517760 n / 2 + 575125383831656140800) a(n - 2) / (n (3 n - 1) (3 n - 2) %1 / 2 13 (2 n - 1) ) - 1/3 (2 n - 5) (139324758020516484 n 12 11 - 2598074019947687112 n + 21467508676826505741 n 10 9 - 103775491836805587738 n + 326398842448508123199 n 8 7 - 702791655839154599876 n + 1061292148279194220834 n 6 5 - 1132460824865511151318 n + 848937553019285355871 n 4 3 - 438678803033019022468 n + 150867112811963954839 n 2 - 32545165522154817560 n + 3957146162230448808 n - 209303570148739704) 2 / 2 2 (n - 2) a(n - 3) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) + 1331/9 / 10 9 (2 n - 5) (2 n - 7) (67176836075466 n - 681684463246527 n 8 7 6 + 2962236507130380 n - 7235043050875940 n + 10946358262282208 n 5 4 3 - 10653200765574773 n + 6705118693140104 n - 2675288529923864 n 2 2 2 + 644547097664306 n - 85054613015232 n + 4798977647112) (n - 2) (n - 3) / 2 2 a(n - 4) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) / 10 9 8 %1 := 67176836075466 n - 1353452824001187 n + 12120354299745093 n 7 6 5 - 63534796113849872 n + 215902912306620556 n - 496973303867043875 n 4 3 2 + 784748936679941851 n - 839389402625599538 n + 582070631004573694 n - 236314764515390988 n + 42660507796575912 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.02853538412124960237390247368629936895188287163175972601908296970486998164\ 204179134736664298217485945 This constant is identified as, 2/77 ln(3) The implied delta is, -0.9051242257588536981775154066983840261979792184970435\ 875644955098693670186036253332210795546429655589 Since this is negative, there is no Apery-style irrationality proof of, 2/77 ln(3), but still a very fast way to compute it to many digits ----------------------- This took, 2.019, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n + k, k) / ----- k By Shalosh B. Ekhad Let ----- \ b(n) = ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n + k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 12 11 10 b(n) = 1/10 (146609894904 n - 2976954986412 n + 26737231598396 n 9 8 7 - 140032754221253 n + 474720390285659 n - 1093280114042732 n 6 5 4 + 1746471943893718 n - 1940797725240509 n + 1481612029378463 n 3 2 - 753938165607294 n + 241627762274060 n - 43637328207000 n / 2 2 13 + 3355812180000) b(n - 1) / (%1 (2 n - 1) n ) - 1/20 (33879178168 n / 12 11 10 - 772624148824 n + 7925437171289 n - 48338075300090 n 9 8 7 + 195183073054951 n - 549627220011524 n + 1107012917787007 n 6 5 4 - 1608360572610690 n + 1677195411687081 n - 1231916888065592 n 3 2 + 614539448902384 n - 195272952864560 n + 35074647702000 n / 2 2 - 2674130760000) b(n - 2) / ((2 n - 3) %1 (2 n - 1) n ) + 1/80 ( / 12 11 10 74429256462 n - 1660165415535 n + 16494695633185 n 9 8 7 - 96248879340791 n + 366091311871193 n - 952244790127157 n 6 5 4 + 1728075255189639 n - 2190887968815701 n + 1911393380828861 n 3 2 - 1108948529100816 n + 402014168020660 n - 81015106176000 n 2 / 2 + 6841467360000) (n - 2) b(n - 3) / (n (2 n - 3) (2 n - 5) %1 / 2 8 7 6 (2 n - 1) ) - 1/20 (74953934 n - 772421287 n + 3311900009 n 5 4 3 2 - 7662423905 n + 10373271269 n - 8328085804 n + 3829612312 n 2 2 2 / 2 - 915516000 n + 87372000) (2 n - 7) (n - 2) (n - 3) b(n - 4) / (n / 2 (2 n - 3) (2 n - 5) %1 (2 n - 1) ) 8 7 6 5 %1 := 74953934 n - 1372052759 n + 10817559170 n - 47952091290 n 4 3 2 + 130645411354 n - 223915575459 n + 235675793702 n - 139242183172 n + 35355556520 and in Maple notation b(n) = 1/10*(146609894904*n^12-2976954986412*n^11+26737231598396*n^10-\ 140032754221253*n^9+474720390285659*n^8-1093280114042732*n^7+1746471943893718*n ^6-1940797725240509*n^5+1481612029378463*n^4-753938165607294*n^3+ 241627762274060*n^2-43637328207000*n+3355812180000)/(74953934*n^8-1372052759*n^ 7+10817559170*n^6-47952091290*n^5+130645411354*n^4-223915575459*n^3+ 235675793702*n^2-139242183172*n+35355556520)/(2*n-1)^2/n^2*b(n-1)-1/20*( 33879178168*n^13-772624148824*n^12+7925437171289*n^11-48338075300090*n^10+ 195183073054951*n^9-549627220011524*n^8+1107012917787007*n^7-1608360572610690*n ^6+1677195411687081*n^5-1231916888065592*n^4+614539448902384*n^3-\ 195272952864560*n^2+35074647702000*n-2674130760000)/(2*n-3)/(74953934*n^8-\ 1372052759*n^7+10817559170*n^6-47952091290*n^5+130645411354*n^4-223915575459*n^ 3+235675793702*n^2-139242183172*n+35355556520)/(2*n-1)^2/n^2*b(n-2)+1/80*( 74429256462*n^12-1660165415535*n^11+16494695633185*n^10-96248879340791*n^9+ 366091311871193*n^8-952244790127157*n^7+1728075255189639*n^6-2190887968815701*n ^5+1911393380828861*n^4-1108948529100816*n^3+402014168020660*n^2-81015106176000 *n+6841467360000)*(n-2)^2/n^2/(2*n-3)/(2*n-5)/(74953934*n^8-1372052759*n^7+ 10817559170*n^6-47952091290*n^5+130645411354*n^4-223915575459*n^3+235675793702* n^2-139242183172*n+35355556520)/(2*n-1)^2*b(n-3)-1/20*(74953934*n^8-772421287*n ^7+3311900009*n^6-7662423905*n^5+10373271269*n^4-8328085804*n^3+3829612312*n^2-\ 915516000*n+87372000)*(2*n-7)^2*(n-2)^2*(n-3)^2/n^2/(2*n-3)/(2*n-5)/(74953934*n ^8-1372052759*n^7+10817559170*n^6-47952091290*n^5+130645411354*n^4-223915575459 *n^3+235675793702*n^2-139242183172*n+35355556520)/(2*n-1)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 10, b(2) = 266, b(3) = 8926 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 12 11 10 a(n) = 1/10 (146609894904 n - 2976954986412 n + 26737231598396 n 9 8 7 - 140032754221253 n + 474720390285659 n - 1093280114042732 n 6 5 4 + 1746471943893718 n - 1940797725240509 n + 1481612029378463 n 3 2 - 753938165607294 n + 241627762274060 n - 43637328207000 n / 2 2 13 + 3355812180000) a(n - 1) / (%1 (2 n - 1) n ) - 1/20 (33879178168 n / 12 11 10 - 772624148824 n + 7925437171289 n - 48338075300090 n 9 8 7 + 195183073054951 n - 549627220011524 n + 1107012917787007 n 6 5 4 - 1608360572610690 n + 1677195411687081 n - 1231916888065592 n 3 2 + 614539448902384 n - 195272952864560 n + 35074647702000 n / 2 2 - 2674130760000) a(n - 2) / ((2 n - 3) %1 (2 n - 1) n ) + 1/80 ( / 12 11 10 74429256462 n - 1660165415535 n + 16494695633185 n 9 8 7 - 96248879340791 n + 366091311871193 n - 952244790127157 n 6 5 4 + 1728075255189639 n - 2190887968815701 n + 1911393380828861 n 3 2 - 1108948529100816 n + 402014168020660 n - 81015106176000 n 2 / 2 + 6841467360000) (n - 2) a(n - 3) / (n (2 n - 3) (2 n - 5) %1 / 2 8 7 6 (2 n - 1) ) - 1/20 (74953934 n - 772421287 n + 3311900009 n 5 4 3 2 - 7662423905 n + 10373271269 n - 8328085804 n + 3829612312 n 2 2 2 / 2 - 915516000 n + 87372000) (2 n - 7) (n - 2) (n - 3) a(n - 4) / (n / 2 (2 n - 3) (2 n - 5) %1 (2 n - 1) ) 8 7 6 5 %1 := 74953934 n - 1372052759 n + 10817559170 n - 47952091290 n 4 3 2 + 130645411354 n - 223915575459 n + 235675793702 n - 139242183172 n + 35355556520 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.10009510254976838113707749603502451754059783314890457087372234976571191057\ 23562700840675034179581292 The implied delta is, -0.2000392997775032139315850967256430953307581487591209\ 029643385253616599571310639785806791087354222165 Since this is negative, there is no Apery-style irrationality proof of, 0.100\ 095102549768381137077496035024517540597833148904570873722349765711910572\ 3562700840675034179581292, but still a very fast way to compute it to many digits ----------------------- This took, 4.889, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n + k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n + k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 12 11 10 b(n) = 1/72 (47834432977604 n - 972161612084754 n + 8739541335993072 n 9 8 7 - 45816627284626437 n + 155477386044656910 n - 358437401719164468 n 6 5 4 + 573217795495357144 n - 637752815478902313 n + 487508449682062278 n 3 2 - 248458816223792964 n + 79776426498352920 n - 14440283708980032 n / 2 2 + 1113517795057920) b(n - 1) / (%1 (2 n - 1) n ) - 1/72 ( / 13 12 11 2433123423904 n - 55532316852064 n + 570098492417310 n 10 9 8 - 3479771806465888 n + 14060664861898489 n - 39617097156660195 n 7 6 5 + 79826057506349357 n - 115998726859960721 n + 120947049654877752 n 4 3 2 - 88786203982344152 n + 44237933676423152 n - 14027167936588512 n / + 2510880103460448 n - 190442160478080) b(n - 2) / ((2 n - 3) %1 / 2 2 12 11 (2 n - 1) n ) + 1/72 (1492573545444 n - 33319419157482 n 10 9 8 + 331355554591778 n - 1935516719252353 n + 7370314706130691 n 7 6 5 - 19194660438581279 n + 34879030334876266 n - 44281303042907300 n 4 3 2 + 38687455529113202 n - 22478378940225658 n + 8160846501620175 n 2 / 2 - 1647037436844924 n + 139291398595440) (n - 2) b(n - 3) / (n / 2 8 7 (2 n - 5) (2 n - 3) %1 (2 n - 1) ) - 1/72 (1758037156 n - 18149041946 n 6 5 4 3 + 77973078562 n - 180796907885 n + 245328717475 n - 197406886169 n 2 2 2 + 90954584787 n - 21774620556 n + 2079866160) (2 n - 7) (n - 2) 2 / 2 2 (n - 3) b(n - 4) / (n (2 n - 5) (2 n - 3) %1 (2 n - 1) ) / 8 7 6 5 %1 := 1758037156 n - 32213339194 n + 254241412552 n - 1128215340859 n 4 3 2 + 3077188504360 n - 5279818955005 n + 5563067726658 n - 3290149920204 n + 836221740696 and in Maple notation b(n) = 1/72*(47834432977604*n^12-972161612084754*n^11+8739541335993072*n^10-\ 45816627284626437*n^9+155477386044656910*n^8-358437401719164468*n^7+ 573217795495357144*n^6-637752815478902313*n^5+487508449682062278*n^4-\ 248458816223792964*n^3+79776426498352920*n^2-14440283708980032*n+ 1113517795057920)/(1758037156*n^8-32213339194*n^7+254241412552*n^6-\ 1128215340859*n^5+3077188504360*n^4-5279818955005*n^3+5563067726658*n^2-\ 3290149920204*n+836221740696)/(2*n-1)^2/n^2*b(n-1)-1/72*(2433123423904*n^13-\ 55532316852064*n^12+570098492417310*n^11-3479771806465888*n^10+ 14060664861898489*n^9-39617097156660195*n^8+79826057506349357*n^7-\ 115998726859960721*n^6+120947049654877752*n^5-88786203982344152*n^4+ 44237933676423152*n^3-14027167936588512*n^2+2510880103460448*n-190442160478080) /(2*n-3)/(1758037156*n^8-32213339194*n^7+254241412552*n^6-1128215340859*n^5+ 3077188504360*n^4-5279818955005*n^3+5563067726658*n^2-3290149920204*n+ 836221740696)/(2*n-1)^2/n^2*b(n-2)+1/72*(1492573545444*n^12-33319419157482*n^11 +331355554591778*n^10-1935516719252353*n^9+7370314706130691*n^8-\ 19194660438581279*n^7+34879030334876266*n^6-44281303042907300*n^5+ 38687455529113202*n^4-22478378940225658*n^3+8160846501620175*n^2-\ 1647037436844924*n+139291398595440)*(n-2)^2/n^2/(2*n-5)/(2*n-3)/(1758037156*n^8 -32213339194*n^7+254241412552*n^6-1128215340859*n^5+3077188504360*n^4-\ 5279818955005*n^3+5563067726658*n^2-3290149920204*n+836221740696)/(2*n-1)^2*b(n -3)-1/72*(1758037156*n^8-18149041946*n^7+77973078562*n^6-180796907885*n^5+ 245328717475*n^4-197406886169*n^3+90954584787*n^2-21774620556*n+2079866160)*(2* n-7)^2*(n-2)^2*(n-3)^2/n^2/(2*n-5)/(2*n-3)/(1758037156*n^8-32213339194*n^7+ 254241412552*n^6-1128215340859*n^5+3077188504360*n^4-5279818955005*n^3+ 5563067726658*n^2-3290149920204*n+836221740696)/(2*n-1)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 19, b(2) = 981, b(3) = 63715 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 12 11 10 a(n) = 1/72 (47834432977604 n - 972161612084754 n + 8739541335993072 n 9 8 7 - 45816627284626437 n + 155477386044656910 n - 358437401719164468 n 6 5 4 + 573217795495357144 n - 637752815478902313 n + 487508449682062278 n 3 2 - 248458816223792964 n + 79776426498352920 n - 14440283708980032 n / 2 2 + 1113517795057920) a(n - 1) / (%1 (2 n - 1) n ) - 1/72 ( / 13 12 11 2433123423904 n - 55532316852064 n + 570098492417310 n 10 9 8 - 3479771806465888 n + 14060664861898489 n - 39617097156660195 n 7 6 5 + 79826057506349357 n - 115998726859960721 n + 120947049654877752 n 4 3 2 - 88786203982344152 n + 44237933676423152 n - 14027167936588512 n / + 2510880103460448 n - 190442160478080) a(n - 2) / ((2 n - 3) %1 / 2 2 12 11 (2 n - 1) n ) + 1/72 (1492573545444 n - 33319419157482 n 10 9 8 + 331355554591778 n - 1935516719252353 n + 7370314706130691 n 7 6 5 - 19194660438581279 n + 34879030334876266 n - 44281303042907300 n 4 3 2 + 38687455529113202 n - 22478378940225658 n + 8160846501620175 n 2 / 2 - 1647037436844924 n + 139291398595440) (n - 2) a(n - 3) / (n / 2 8 7 (2 n - 5) (2 n - 3) %1 (2 n - 1) ) - 1/72 (1758037156 n - 18149041946 n 6 5 4 3 + 77973078562 n - 180796907885 n + 245328717475 n - 197406886169 n 2 2 2 + 90954584787 n - 21774620556 n + 2079866160) (2 n - 7) (n - 2) 2 / 2 2 (n - 3) a(n - 4) / (n (2 n - 5) (2 n - 3) %1 (2 n - 1) ) / 8 7 6 5 %1 := 1758037156 n - 32213339194 n + 254241412552 n - 1128215340859 n 4 3 2 + 3077188504360 n - 5279818955005 n + 5563067726658 n - 3290149920204 n + 836221740696 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.05264408966278065641143535099682353681586076969824723449017822856911850293\ 440719359031873369593053323 This constant is identified as, 6/79 ln(2) The implied delta is, -0.1997716735644068679762539748894298709925409479812602\ 288235781049042999046797502816990097658494464349 Since this is negative, there is no Apery-style irrationality proof of, 6/79 ln(2), but still a very fast way to compute it to many digits ----------------------- This took, 2.583, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n + k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n + k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 12 11 b(n) = 1/78 (520640472222444 n - 10583457095988714 n 10 9 8 + 95164505402148540 n - 499013534581291009 n + 1693813164797626561 n 7 6 5 - 3905966433389012268 n + 6248314802292447228 n - 6954092579437103727 n 4 3 2 + 5317904651700860943 n - 2711568621226108346 n + 871165703511378020 n / 2 2 - 157808001427639992 n + 12180063967737120) b(n - 1) / (%1 (2 n - 1) n / 13 12 11 ) - 1/156 (33387526964052 n - 762162614795592 n + 7825823592520497 n 10 9 8 - 47774676720609192 n + 193063365520685811 n - 543995269053870884 n 7 6 5 + 1096067332392412011 n - 1592485357874750244 n + 1659894514265326869 n 4 3 2 - 1217876229998049672 n + 606314197588649928 n - 192014520553773136 n / + 34307195121577872 n - 2595172672048320) b(n - 2) / ((2 n - 3) %1 / 2 2 12 11 (2 n - 1) n ) + 1/624 (86471550027882 n - 1930716436110831 n 10 9 8 + 19204998656010513 n - 112209823311119685 n + 427413544695521739 n 7 6 5 - 1113485528747432189 n + 2024058650501395355 n - 2570663152639215083 n 4 3 2 + 2246835980698290611 n - 1306026061642028372 n + 474369242184320300 n 2 / 2 - 95783134562378640 n + 8104424158958400) (n - 2) b(n - 3) / (n / 2 8 (2 n - 3) (2 n - 5) %1 (2 n - 1) ) - 1/156 (11915605626 n 7 6 5 4 - 123061573071 n + 528956544033 n - 1227146075603 n + 1666090416651 n 3 2 - 1341394923786 n + 618359513870 n - 148098174120 n + 14150635200) 2 2 2 / 2 (2 n - 7) (n - 2) (n - 3) b(n - 4) / (n (2 n - 3) (2 n - 5) %1 / 2 (2 n - 1) ) 8 7 6 5 %1 := 11915605626 n - 218386418079 n + 1724024513058 n - 7652452289348 n 4 3 2 + 20877416406466 n - 35830777199621 n + 37762825693678 n - 22339589138540 n + 5679173461960 and in Maple notation b(n) = 1/78*(520640472222444*n^12-10583457095988714*n^11+95164505402148540*n^10 -499013534581291009*n^9+1693813164797626561*n^8-3905966433389012268*n^7+ 6248314802292447228*n^6-6954092579437103727*n^5+5317904651700860943*n^4-\ 2711568621226108346*n^3+871165703511378020*n^2-157808001427639992*n+ 12180063967737120)/(11915605626*n^8-218386418079*n^7+1724024513058*n^6-\ 7652452289348*n^5+20877416406466*n^4-35830777199621*n^3+37762825693678*n^2-\ 22339589138540*n+5679173461960)/(2*n-1)^2/n^2*b(n-1)-1/156*(33387526964052*n^13 -762162614795592*n^12+7825823592520497*n^11-47774676720609192*n^10+ 193063365520685811*n^9-543995269053870884*n^8+1096067332392412011*n^7-\ 1592485357874750244*n^6+1659894514265326869*n^5-1217876229998049672*n^4+ 606314197588649928*n^3-192014520553773136*n^2+34307195121577872*n-\ 2595172672048320)/(2*n-3)/(11915605626*n^8-218386418079*n^7+1724024513058*n^6-\ 7652452289348*n^5+20877416406466*n^4-35830777199621*n^3+37762825693678*n^2-\ 22339589138540*n+5679173461960)/(2*n-1)^2/n^2*b(n-2)+1/624*(86471550027882*n^12 -1930716436110831*n^11+19204998656010513*n^10-112209823311119685*n^9+ 427413544695521739*n^8-1113485528747432189*n^7+2024058650501395355*n^6-\ 2570663152639215083*n^5+2246835980698290611*n^4-1306026061642028372*n^3+ 474369242184320300*n^2-95783134562378640*n+8104424158958400)*(n-2)^2/n^2/(2*n-3 )/(2*n-5)/(11915605626*n^8-218386418079*n^7+1724024513058*n^6-7652452289348*n^5 +20877416406466*n^4-35830777199621*n^3+37762825693678*n^2-22339589138540*n+ 5679173461960)/(2*n-1)^2*b(n-3)-1/156*(11915605626*n^8-123061573071*n^7+ 528956544033*n^6-1227146075603*n^5+1666090416651*n^4-1341394923786*n^3+ 618359513870*n^2-148098174120*n+14150635200)*(2*n-7)^2*(n-2)^2*(n-3)^2/n^2/(2*n -3)/(2*n-5)/(11915605626*n^8-218386418079*n^7+1724024513058*n^6-7652452289348*n ^5+20877416406466*n^4-35830777199621*n^3+37762825693678*n^2-22339589138540*n+ 5679173461960)/(2*n-1)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 28, b(2) = 2146, b(3) = 206704 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 12 11 a(n) = 1/78 (520640472222444 n - 10583457095988714 n 10 9 8 + 95164505402148540 n - 499013534581291009 n + 1693813164797626561 n 7 6 5 - 3905966433389012268 n + 6248314802292447228 n - 6954092579437103727 n 4 3 2 + 5317904651700860943 n - 2711568621226108346 n + 871165703511378020 n / 2 2 - 157808001427639992 n + 12180063967737120) a(n - 1) / (%1 (2 n - 1) n / 13 12 11 ) - 1/156 (33387526964052 n - 762162614795592 n + 7825823592520497 n 10 9 8 - 47774676720609192 n + 193063365520685811 n - 543995269053870884 n 7 6 5 + 1096067332392412011 n - 1592485357874750244 n + 1659894514265326869 n 4 3 2 - 1217876229998049672 n + 606314197588649928 n - 192014520553773136 n / + 34307195121577872 n - 2595172672048320) a(n - 2) / ((2 n - 3) %1 / 2 2 12 11 (2 n - 1) n ) + 1/624 (86471550027882 n - 1930716436110831 n 10 9 8 + 19204998656010513 n - 112209823311119685 n + 427413544695521739 n 7 6 5 - 1113485528747432189 n + 2024058650501395355 n - 2570663152639215083 n 4 3 2 + 2246835980698290611 n - 1306026061642028372 n + 474369242184320300 n 2 / 2 - 95783134562378640 n + 8104424158958400) (n - 2) a(n - 3) / (n / 2 8 (2 n - 3) (2 n - 5) %1 (2 n - 1) ) - 1/156 (11915605626 n 7 6 5 4 - 123061573071 n + 528956544033 n - 1227146075603 n + 1666090416651 n 3 2 - 1341394923786 n + 618359513870 n - 148098174120 n + 14150635200) 2 2 2 / 2 (2 n - 7) (n - 2) (n - 3) a(n - 4) / (n (2 n - 3) (2 n - 5) %1 / 2 (2 n - 1) ) 8 7 6 5 %1 := 11915605626 n - 218386418079 n + 1724024513058 n - 7652452289348 n 4 3 2 + 20877416406466 n - 35830777199621 n + 37762825693678 n - 22339589138540 n + 5679173461960 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.03571803942808124016883688626862374001327701462411091014323961646474918089\ 830332313779471632847152211 The implied delta is, -0.2012224203519101370334405816119339573371475647225492\ 362646421198527160382688645591295778640910171385 Since this is negative, there is no Apery-style irrationality proof of, 0.035\ 718039428081240168836886268623740013277014624110910143239616464749180898\ 30332313779471632847152211, but still a very fast way to compute it to many digits ----------------------- This took, 5.507, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n + 2 k, k) / ----- k By Shalosh B. Ekhad Let ----- \ b(n) = ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n + 2 k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 16 15 14 b(n) = 2/9 (33762201421312 n - 816920307486976 n + 8997390452139152 n 13 12 11 - 59796865467273916 n + 268037933923322652 n - 857984624899860028 n 10 9 + 2025523340108361302 n - 3591252667466616947 n 8 7 6 + 4823625073753396606 n - 4914401046255353108 n + 3776861223446589510 n 5 4 3 - 2161473025663642715 n + 900863361811070886 n - 263724933147858102 n 2 + 51038748826056612 n - 5826058176273120 n + 295365716956800) b(n - 1) / 2 2 2 15 / (n %1 (3 n - 1) (3 n - 2) ) - 4/9 (n - 1) (5804952554336 n / 14 13 12 - 140458365450128 n + 1541135769882768 n - 10152843084969084 n 11 10 9 + 44815804713023620 n - 140053878874610312 n + 319196237263316094 n 8 7 6 - 538467183774724759 n + 675374547608327616 n - 627246299635751736 n 5 4 3 + 426000872535796229 n - 206899294881547855 n + 69302107706713429 n 2 - 15071642696443602 n + 1899929455434144 n - 104599802926080) b(n - 2) / 2 2 2 14 / (n %1 (3 n - 1) (3 n - 2) ) + 4/9 (n - 1) (n - 2) (647145734976 n / 13 12 11 - 15011385715872 n + 156391937548848 n - 967290375920376 n 10 9 8 + 3955739396601012 n - 11275235914709756 n + 23008986795620672 n 7 6 5 - 33999849652639084 n + 36384702082358675 n - 27923993590659244 n 4 3 2 + 15058261062908363 n - 5511002105222216 n + 1290536221144506 n / 2 2 - 172945025878464 n + 10015069463280) b(n - 3) / (n %1 (3 n - 1) / 2 10 9 (3 n - 2) ) - 68/9 (n - 1) (n - 2) (1382790032 n - 15482130120 n 8 7 6 5 + 74002100540 n - 197832708140 n + 325904041804 n - 344111505362 n 4 3 2 + 235099547825 n - 102539756111 n + 27356653436 n - 4040740584 n 2 2 / 2 2 2 + 251516880) (n - 3) (2 n - 7) b(n - 4) / (n %1 (3 n - 1) (3 n - 2) / ) 10 9 8 7 %1 := 1382790032 n - 29310030440 n + 275566823060 n - 1513141000620 n 6 5 4 + 5373676650704 n - 12897351740550 n + 21189643826235 n 3 2 - 23536829986171 n + 16921376811219 n - 7112766117423 n + 1328003490834 and in Maple notation b(n) = 2/9*(33762201421312*n^16-816920307486976*n^15+8997390452139152*n^14-\ 59796865467273916*n^13+268037933923322652*n^12-857984624899860028*n^11+ 2025523340108361302*n^10-3591252667466616947*n^9+4823625073753396606*n^8-\ 4914401046255353108*n^7+3776861223446589510*n^6-2161473025663642715*n^5+ 900863361811070886*n^4-263724933147858102*n^3+51038748826056612*n^2-\ 5826058176273120*n+295365716956800)/n^2/(1382790032*n^10-29310030440*n^9+ 275566823060*n^8-1513141000620*n^7+5373676650704*n^6-12897351740550*n^5+ 21189643826235*n^4-23536829986171*n^3+16921376811219*n^2-7112766117423*n+ 1328003490834)/(3*n-1)^2/(3*n-2)^2*b(n-1)-4/9*(n-1)*(5804952554336*n^15-\ 140458365450128*n^14+1541135769882768*n^13-10152843084969084*n^12+ 44815804713023620*n^11-140053878874610312*n^10+319196237263316094*n^9-\ 538467183774724759*n^8+675374547608327616*n^7-627246299635751736*n^6+ 426000872535796229*n^5-206899294881547855*n^4+69302107706713429*n^3-\ 15071642696443602*n^2+1899929455434144*n-104599802926080)/n^2/(1382790032*n^10-\ 29310030440*n^9+275566823060*n^8-1513141000620*n^7+5373676650704*n^6-\ 12897351740550*n^5+21189643826235*n^4-23536829986171*n^3+16921376811219*n^2-\ 7112766117423*n+1328003490834)/(3*n-1)^2/(3*n-2)^2*b(n-2)+4/9*(n-1)*(n-2)*( 647145734976*n^14-15011385715872*n^13+156391937548848*n^12-967290375920376*n^11 +3955739396601012*n^10-11275235914709756*n^9+23008986795620672*n^8-\ 33999849652639084*n^7+36384702082358675*n^6-27923993590659244*n^5+ 15058261062908363*n^4-5511002105222216*n^3+1290536221144506*n^2-172945025878464 *n+10015069463280)/n^2/(1382790032*n^10-29310030440*n^9+275566823060*n^8-\ 1513141000620*n^7+5373676650704*n^6-12897351740550*n^5+21189643826235*n^4-\ 23536829986171*n^3+16921376811219*n^2-7112766117423*n+1328003490834)/(3*n-1)^2/ (3*n-2)^2*b(n-3)-68/9*(n-1)*(n-2)*(1382790032*n^10-15482130120*n^9+74002100540* n^8-197832708140*n^7+325904041804*n^6-344111505362*n^5+235099547825*n^4-\ 102539756111*n^3+27356653436*n^2-4040740584*n+251516880)*(n-3)^2*(2*n-7)^2/n^2/ (1382790032*n^10-29310030440*n^9+275566823060*n^8-1513141000620*n^7+ 5373676650704*n^6-12897351740550*n^5+21189643826235*n^4-23536829986171*n^3+ 16921376811219*n^2-7112766117423*n+1328003490834)/(3*n-1)^2/(3*n-2)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 13, b(2) = 469, b(3) = 21436 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 16 15 14 a(n) = 2/9 (33762201421312 n - 816920307486976 n + 8997390452139152 n 13 12 11 - 59796865467273916 n + 268037933923322652 n - 857984624899860028 n 10 9 + 2025523340108361302 n - 3591252667466616947 n 8 7 6 + 4823625073753396606 n - 4914401046255353108 n + 3776861223446589510 n 5 4 3 - 2161473025663642715 n + 900863361811070886 n - 263724933147858102 n 2 + 51038748826056612 n - 5826058176273120 n + 295365716956800) a(n - 1) / 2 2 2 15 / (n %1 (3 n - 1) (3 n - 2) ) - 4/9 (n - 1) (5804952554336 n / 14 13 12 - 140458365450128 n + 1541135769882768 n - 10152843084969084 n 11 10 9 + 44815804713023620 n - 140053878874610312 n + 319196237263316094 n 8 7 6 - 538467183774724759 n + 675374547608327616 n - 627246299635751736 n 5 4 3 + 426000872535796229 n - 206899294881547855 n + 69302107706713429 n 2 - 15071642696443602 n + 1899929455434144 n - 104599802926080) a(n - 2) / 2 2 2 14 / (n %1 (3 n - 1) (3 n - 2) ) + 4/9 (n - 1) (n - 2) (647145734976 n / 13 12 11 - 15011385715872 n + 156391937548848 n - 967290375920376 n 10 9 8 + 3955739396601012 n - 11275235914709756 n + 23008986795620672 n 7 6 5 - 33999849652639084 n + 36384702082358675 n - 27923993590659244 n 4 3 2 + 15058261062908363 n - 5511002105222216 n + 1290536221144506 n / 2 2 - 172945025878464 n + 10015069463280) a(n - 3) / (n %1 (3 n - 1) / 2 10 9 (3 n - 2) ) - 68/9 (n - 1) (n - 2) (1382790032 n - 15482130120 n 8 7 6 5 + 74002100540 n - 197832708140 n + 325904041804 n - 344111505362 n 4 3 2 + 235099547825 n - 102539756111 n + 27356653436 n - 4040740584 n 2 2 / 2 2 2 + 251516880) (n - 3) (2 n - 7) a(n - 4) / (n %1 (3 n - 1) (3 n - 2) / ) 10 9 8 7 %1 := 1382790032 n - 29310030440 n + 275566823060 n - 1513141000620 n 6 5 4 + 5373676650704 n - 12897351740550 n + 21189643826235 n 3 2 - 23536829986171 n + 16921376811219 n - 7112766117423 n + 1328003490834 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.07708381652240925625474442639460593147242390560621096547108088786325613345\ 282431212802677138601275334 The implied delta is, -0.4618025872883926827040984707277121654795099380869658\ 832865755541673641771037504863577383976009771478 Since this is negative, there is no Apery-style irrationality proof of, 0.077\ 083816522409256254744426394605931472423905606210965471080887863256133452\ 82431212802677138601275334, but still a very fast way to compute it to many digits ----------------------- This took, 7.271, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n + 2 k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n + 2 k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 16 15 b(n) = 1/9 (2931875264071936 n - 70996727248859776 n 14 13 + 782900977718382368 n - 5211515988363274464 n 12 11 + 23405375851066844184 n - 75083611632941438372 n 10 9 + 177679305835463788412 n - 315823215249078760256 n 8 7 + 425318523264177504823 n - 434491419483972382887 n 6 5 + 334830686919897012492 n - 192145780457661350278 n 4 3 + 80300843103331008567 n - 23570686017671866815 n 2 + 4573356190687976706 n - 523272726475626240 n + 26579699543577600) / 2 2 2 b(n - 1) / (n %1 (3 n - 1) (3 n - 2) ) - 1/9 (n - 1) ( / 15 14 13 490162520031488 n - 11869514084974208 n + 130403114453252608 n 12 11 - 860672203341422080 n + 3808472004513144768 n 10 9 - 11939233843082047144 n + 27316043028174788952 n 8 7 - 46295069991652855354 n + 58382239736212753991 n 6 5 - 54558880050359773391 n + 37308353414302913736 n 4 3 - 18251034604924906721 n + 6157371972444209401 n 2 - 1348095120913792686 n + 170965070434992960 n - 9465504716908800) / 2 2 2 b(n - 2) / (n %1 (3 n - 1) (3 n - 2) ) + 1/9 (n - 1) (n - 2) ( / 14 13 12 44146599270272 n - 1024883896591680 n + 10694771982733472 n 11 10 9 - 66310230705881488 n + 272082809620288800 n - 778844130747997408 n 8 7 6 + 1597669829524362604 n - 2375416596393632034 n + 2559939427023353143 n 5 4 3 - 1979839623091932020 n + 1076253224937873915 n - 396995372940654626 n 2 + 93633986593513770 n - 12625259978931840 n + 734962256683200) b(n - 3) / 2 2 2 10 / (n %1 (3 n - 1) (3 n - 2) ) - 11/3 (n - 1) (n - 2) (30742757152 n / 9 8 7 6 - 344794363056 n + 1654534091920 n - 4450335700224 n + 7391714354316 n 5 4 3 2 - 7882409508282 n + 5444433020297 n - 2400415331703 n + 646576538972 n 2 2 / 2 - 96277425696 n + 6037485120) (n - 3) (2 n - 7) b(n - 4) / (n %1 / 2 2 (3 n - 1) (3 n - 2) ) 10 9 8 7 %1 := 30742757152 n - 652221934576 n + 6141107431264 n - 33788336363840 n 6 5 4 + 120289724328268 n - 289534919033762 n + 477211400565663 n 3 2 - 531904045592335 n + 383794256473743 n - 161929941723195 n + 30348270576738 and in Maple notation b(n) = 1/9*(2931875264071936*n^16-70996727248859776*n^15+782900977718382368*n^ 14-5211515988363274464*n^13+23405375851066844184*n^12-75083611632941438372*n^11 +177679305835463788412*n^10-315823215249078760256*n^9+425318523264177504823*n^8 -434491419483972382887*n^7+334830686919897012492*n^6-192145780457661350278*n^5+ 80300843103331008567*n^4-23570686017671866815*n^3+4573356190687976706*n^2-\ 523272726475626240*n+26579699543577600)/n^2/(30742757152*n^10-652221934576*n^9+ 6141107431264*n^8-33788336363840*n^7+120289724328268*n^6-289534919033762*n^5+ 477211400565663*n^4-531904045592335*n^3+383794256473743*n^2-161929941723195*n+ 30348270576738)/(3*n-1)^2/(3*n-2)^2*b(n-1)-1/9*(n-1)*(490162520031488*n^15-\ 11869514084974208*n^14+130403114453252608*n^13-860672203341422080*n^12+ 3808472004513144768*n^11-11939233843082047144*n^10+27316043028174788952*n^9-\ 46295069991652855354*n^8+58382239736212753991*n^7-54558880050359773391*n^6+ 37308353414302913736*n^5-18251034604924906721*n^4+6157371972444209401*n^3-\ 1348095120913792686*n^2+170965070434992960*n-9465504716908800)/n^2/(30742757152 *n^10-652221934576*n^9+6141107431264*n^8-33788336363840*n^7+120289724328268*n^6 -289534919033762*n^5+477211400565663*n^4-531904045592335*n^3+383794256473743*n^ 2-161929941723195*n+30348270576738)/(3*n-1)^2/(3*n-2)^2*b(n-2)+1/9*(n-1)*(n-2)* (44146599270272*n^14-1024883896591680*n^13+10694771982733472*n^12-\ 66310230705881488*n^11+272082809620288800*n^10-778844130747997408*n^9+ 1597669829524362604*n^8-2375416596393632034*n^7+2559939427023353143*n^6-\ 1979839623091932020*n^5+1076253224937873915*n^4-396995372940654626*n^3+ 93633986593513770*n^2-12625259978931840*n+734962256683200)/n^2/(30742757152*n^ 10-652221934576*n^9+6141107431264*n^8-33788336363840*n^7+120289724328268*n^6-\ 289534919033762*n^5+477211400565663*n^4-531904045592335*n^3+383794256473743*n^2 -161929941723195*n+30348270576738)/(3*n-1)^2/(3*n-2)^2*b(n-3)-11/3*(n-1)*(n-2)* (30742757152*n^10-344794363056*n^9+1654534091920*n^8-4450335700224*n^7+ 7391714354316*n^6-7882409508282*n^5+5444433020297*n^4-2400415331703*n^3+ 646576538972*n^2-96277425696*n+6037485120)*(n-3)^2*(2*n-7)^2/n^2/(30742757152*n ^10-652221934576*n^9+6141107431264*n^8-33788336363840*n^7+120289724328268*n^6-\ 289534919033762*n^5+477211400565663*n^4-531904045592335*n^3+383794256473743*n^2 -161929941723195*n+30348270576738)/(3*n-1)^2/(3*n-2)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 25, b(2) = 1777, b(3) = 159421 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 16 15 a(n) = 1/9 (2931875264071936 n - 70996727248859776 n 14 13 + 782900977718382368 n - 5211515988363274464 n 12 11 + 23405375851066844184 n - 75083611632941438372 n 10 9 + 177679305835463788412 n - 315823215249078760256 n 8 7 + 425318523264177504823 n - 434491419483972382887 n 6 5 + 334830686919897012492 n - 192145780457661350278 n 4 3 + 80300843103331008567 n - 23570686017671866815 n 2 + 4573356190687976706 n - 523272726475626240 n + 26579699543577600) / 2 2 2 a(n - 1) / (n %1 (3 n - 1) (3 n - 2) ) - 1/9 (n - 1) ( / 15 14 13 490162520031488 n - 11869514084974208 n + 130403114453252608 n 12 11 - 860672203341422080 n + 3808472004513144768 n 10 9 - 11939233843082047144 n + 27316043028174788952 n 8 7 - 46295069991652855354 n + 58382239736212753991 n 6 5 - 54558880050359773391 n + 37308353414302913736 n 4 3 - 18251034604924906721 n + 6157371972444209401 n 2 - 1348095120913792686 n + 170965070434992960 n - 9465504716908800) / 2 2 2 a(n - 2) / (n %1 (3 n - 1) (3 n - 2) ) + 1/9 (n - 1) (n - 2) ( / 14 13 12 44146599270272 n - 1024883896591680 n + 10694771982733472 n 11 10 9 - 66310230705881488 n + 272082809620288800 n - 778844130747997408 n 8 7 6 + 1597669829524362604 n - 2375416596393632034 n + 2559939427023353143 n 5 4 3 - 1979839623091932020 n + 1076253224937873915 n - 396995372940654626 n 2 + 93633986593513770 n - 12625259978931840 n + 734962256683200) a(n - 3) / 2 2 2 10 / (n %1 (3 n - 1) (3 n - 2) ) - 11/3 (n - 1) (n - 2) (30742757152 n / 9 8 7 6 - 344794363056 n + 1654534091920 n - 4450335700224 n + 7391714354316 n 5 4 3 2 - 7882409508282 n + 5444433020297 n - 2400415331703 n + 646576538972 n 2 2 / 2 - 96277425696 n + 6037485120) (n - 3) (2 n - 7) a(n - 4) / (n %1 / 2 2 (3 n - 1) (3 n - 2) ) 10 9 8 7 %1 := 30742757152 n - 652221934576 n + 6141107431264 n - 33788336363840 n 6 5 4 + 120289724328268 n - 289534919033762 n + 477211400565663 n 3 2 - 531904045592335 n + 383794256473743 n - 161929941723195 n + 30348270576738 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.04002885373269341284835049125539513227961373321197292129471648116329789107\ 982057295506735987358884521 The implied delta is, -0.4356198621891237697104486967816099450959577469259498\ 238576048017883463841163905712534418629269865396 Since this is negative, there is no Apery-style irrationality proof of, 0.040\ 028853732693412848350491255395132279613733211972921294716481163297891079\ 82057295506735987358884521, but still a very fast way to compute it to many digits ----------------------- This took, 8.706, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n + 2 k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n + 2 k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 16 15 b(n) = 2/81 (128497239609398400 n - 3112761205396939200 n 14 13 + 34345417290576238800 n - 228803562403738788260 n 12 11 + 1028535770673700906092 n - 3302997028521842942488 n 10 9 + 7825272925840021644042 n - 13926235910289347242131 n 8 7 + 18777894159821846304942 n - 19207136041572044888704 n 6 5 + 14820174797314619699274 n - 8515242416561157904907 n 4 3 + 3562955073218115434190 n - 1047047813800287639030 n 2 + 203378003954398288260 n - 23292999112440123360 n + 1184133976143350400) / 2 2 2 b(n - 1) / (n %1 (3 n - 1) (3 n - 2) ) - 4/81 (n - 1) ( / 15 14 13 7089172775100000 n - 171730552810050000 n + 1887864001985318400 n 12 11 - 12471216714280649820 n + 55251258197116303740 n 10 9 - 173473154113028450800 n + 397642988067484830182 n 8 7 - 675451472451117045855 n + 854066396315000315160 n 6 5 - 800550325162481208888 n + 549260201123119756341 n 4 3 - 269646346685451368015 n + 91294144805520534301 n 2 - 20054748000835577442 n + 2550882268357014816 n - 141591568451397120) / 2 2 2 b(n - 2) / (n %1 (3 n - 1) (3 n - 2) ) + 4/81 (n - 1) (n - 2) ( / 14 13 12 586440505310400 n - 13619695437784800 n + 142231711903750800 n 11 10 - 882882552472392360 n + 3628148568465988572 n 9 8 - 10405371442147524020 n + 21393063306892695308 n 7 6 - 31889587650802556632 n + 34465531413211968071 n 5 4 - 26737365735758218336 n + 14580360549067214051 n 3 2 - 5394625116788678624 n + 1275871055962718202 n - 172435742329318272 n / 2 2 2 196 + 10056781528118640) b(n - 3) / (n %1 (3 n - 1) (3 n - 2) ) - --- / 81 10 9 8 (n - 1) (n - 2) (201111284400 n - 2257342035000 n + 10852926995100 n 7 6 5 - 29280832962820 n + 48831757675652 n - 52329773348458 n 4 3 2 + 36340723184225 n - 16109138683447 n + 4360283292188 n 2 2 / 2 - 651903197544 n + 41018147280) (n - 3) (2 n - 7) b(n - 4) / (n %1 / 2 2 (3 n - 1) (3 n - 2) ) 10 9 8 %1 := 201111284400 n - 4268454879000 n + 40219013108100 n 7 6 5 - 221501916311620 n + 789529644942192 n - 1903086863425990 n 4 3 2 + 3141658464550995 n - 3507748068910267 n + 2535599906207259 n - 1071818629224903 n + 201256810806114 and in Maple notation b(n) = 2/81*(128497239609398400*n^16-3112761205396939200*n^15+ 34345417290576238800*n^14-228803562403738788260*n^13+1028535770673700906092*n^ 12-3302997028521842942488*n^11+7825272925840021644042*n^10-\ 13926235910289347242131*n^9+18777894159821846304942*n^8-19207136041572044888704 *n^7+14820174797314619699274*n^6-8515242416561157904907*n^5+ 3562955073218115434190*n^4-1047047813800287639030*n^3+203378003954398288260*n^2 -23292999112440123360*n+1184133976143350400)/n^2/(201111284400*n^10-\ 4268454879000*n^9+40219013108100*n^8-221501916311620*n^7+789529644942192*n^6-\ 1903086863425990*n^5+3141658464550995*n^4-3507748068910267*n^3+2535599906207259 *n^2-1071818629224903*n+201256810806114)/(3*n-1)^2/(3*n-2)^2*b(n-1)-4/81*(n-1)* (7089172775100000*n^15-171730552810050000*n^14+1887864001985318400*n^13-\ 12471216714280649820*n^12+55251258197116303740*n^11-173473154113028450800*n^10+ 397642988067484830182*n^9-675451472451117045855*n^8+854066396315000315160*n^7-\ 800550325162481208888*n^6+549260201123119756341*n^5-269646346685451368015*n^4+ 91294144805520534301*n^3-20054748000835577442*n^2+2550882268357014816*n-\ 141591568451397120)/n^2/(201111284400*n^10-4268454879000*n^9+40219013108100*n^8 -221501916311620*n^7+789529644942192*n^6-1903086863425990*n^5+3141658464550995* n^4-3507748068910267*n^3+2535599906207259*n^2-1071818629224903*n+ 201256810806114)/(3*n-1)^2/(3*n-2)^2*b(n-2)+4/81*(n-1)*(n-2)*(586440505310400*n ^14-13619695437784800*n^13+142231711903750800*n^12-882882552472392360*n^11+ 3628148568465988572*n^10-10405371442147524020*n^9+21393063306892695308*n^8-\ 31889587650802556632*n^7+34465531413211968071*n^6-26737365735758218336*n^5+ 14580360549067214051*n^4-5394625116788678624*n^3+1275871055962718202*n^2-\ 172435742329318272*n+10056781528118640)/n^2/(201111284400*n^10-4268454879000*n^ 9+40219013108100*n^8-221501916311620*n^7+789529644942192*n^6-1903086863425990*n ^5+3141658464550995*n^4-3507748068910267*n^3+2535599906207259*n^2-\ 1071818629224903*n+201256810806114)/(3*n-1)^2/(3*n-2)^2*b(n-3)-196/81*(n-1)*(n-\ 2)*(201111284400*n^10-2257342035000*n^9+10852926995100*n^8-29280832962820*n^7+ 48831757675652*n^6-52329773348458*n^5+36340723184225*n^4-16109138683447*n^3+ 4360283292188*n^2-651903197544*n+41018147280)*(n-3)^2*(2*n-7)^2/n^2/( 201111284400*n^10-4268454879000*n^9+40219013108100*n^8-221501916311620*n^7+ 789529644942192*n^6-1903086863425990*n^5+3141658464550995*n^4-3507748068910267* n^3+2535599906207259*n^2-1071818629224903*n+201256810806114)/(3*n-1)^2/(3*n-2)^ 2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 37, b(2) = 3925, b(3) = 524836 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 16 15 a(n) = 2/81 (128497239609398400 n - 3112761205396939200 n 14 13 + 34345417290576238800 n - 228803562403738788260 n 12 11 + 1028535770673700906092 n - 3302997028521842942488 n 10 9 + 7825272925840021644042 n - 13926235910289347242131 n 8 7 + 18777894159821846304942 n - 19207136041572044888704 n 6 5 + 14820174797314619699274 n - 8515242416561157904907 n 4 3 + 3562955073218115434190 n - 1047047813800287639030 n 2 + 203378003954398288260 n - 23292999112440123360 n + 1184133976143350400) / 2 2 2 a(n - 1) / (n %1 (3 n - 1) (3 n - 2) ) - 4/81 (n - 1) ( / 15 14 13 7089172775100000 n - 171730552810050000 n + 1887864001985318400 n 12 11 - 12471216714280649820 n + 55251258197116303740 n 10 9 - 173473154113028450800 n + 397642988067484830182 n 8 7 - 675451472451117045855 n + 854066396315000315160 n 6 5 - 800550325162481208888 n + 549260201123119756341 n 4 3 - 269646346685451368015 n + 91294144805520534301 n 2 - 20054748000835577442 n + 2550882268357014816 n - 141591568451397120) / 2 2 2 a(n - 2) / (n %1 (3 n - 1) (3 n - 2) ) + 4/81 (n - 1) (n - 2) ( / 14 13 12 586440505310400 n - 13619695437784800 n + 142231711903750800 n 11 10 - 882882552472392360 n + 3628148568465988572 n 9 8 - 10405371442147524020 n + 21393063306892695308 n 7 6 - 31889587650802556632 n + 34465531413211968071 n 5 4 - 26737365735758218336 n + 14580360549067214051 n 3 2 - 5394625116788678624 n + 1275871055962718202 n - 172435742329318272 n / 2 2 2 196 + 10056781528118640) a(n - 3) / (n %1 (3 n - 1) (3 n - 2) ) - --- / 81 10 9 8 (n - 1) (n - 2) (201111284400 n - 2257342035000 n + 10852926995100 n 7 6 5 - 29280832962820 n + 48831757675652 n - 52329773348458 n 4 3 2 + 36340723184225 n - 16109138683447 n + 4360283292188 n 2 2 / 2 - 651903197544 n + 41018147280) (n - 3) (2 n - 7) a(n - 4) / (n %1 / 2 2 (3 n - 1) (3 n - 2) ) 10 9 8 %1 := 201111284400 n - 4268454879000 n + 40219013108100 n 7 6 5 - 221501916311620 n + 789529644942192 n - 1903086863425990 n 4 3 2 + 3141658464550995 n - 3507748068910267 n + 2535599906207259 n - 1071818629224903 n + 201256810806114 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.02703717235871779214222271421111066382389065620124904781353367693549045402\ 703436274858368021974155964 The implied delta is, -0.4260205806131543151499842738728939964771979530565174\ 743665575856315344565440264349130221045576394809 Since this is negative, there is no Apery-style irrationality proof of, 0.027\ 037172358717792142222714211110663823890656201249047813533676935490454027\ 03436274858368021974155964, but still a very fast way to compute it to many digits ----------------------- This took, 9.092, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 ) binomial(n, k) binomial(2 n, k) / ----- k By Shalosh B. Ekhad Let ----- \ 2 b(n) = ) binomial(n, k) binomial(2 n, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 8 7 6 5 4 b(n) = 1/4 (91390 n - 815480 n + 3029921 n - 6092265 n + 7223517 n 3 2 / 2 - 5167735 n + 2187096 n - 504180 n + 48960) b(n - 1) / (n %1 / 2 (2 n - 1) ) + 2 (2 n - 3) 5 4 3 2 2 (8892 n - 57114 n + 134370 n - 140549 n + 62831 n - 10064) (n - 1) / 2 2 b(n - 2) / (n %1 (2 n - 1) ) + 16 (2 n - 3) (2 n - 5) / 4 3 2 2 / 2 (494 n - 1444 n + 1499 n - 653 n + 102) (n - 2) b(n - 3) / (n %1 / 2 (2 n - 1) ) 4 3 2 %1 := 494 n - 3420 n + 8795 n - 9959 n + 4192 and in Maple notation b(n) = 1/4*(91390*n^8-815480*n^7+3029921*n^6-6092265*n^5+7223517*n^4-5167735*n^ 3+2187096*n^2-504180*n+48960)/n^2/(494*n^4-3420*n^3+8795*n^2-9959*n+4192)/(2*n-\ 1)^2*b(n-1)+2*(2*n-3)*(8892*n^5-57114*n^4+134370*n^3-140549*n^2+62831*n-10064)* (n-1)^2/n^2/(494*n^4-3420*n^3+8795*n^2-9959*n+4192)/(2*n-1)^2*b(n-2)+16*(2*n-3) *(2*n-5)*(494*n^4-1444*n^3+1499*n^2-653*n+102)*(n-2)^2/n^2/(494*n^4-3420*n^3+ 8795*n^2-9959*n+4192)/(2*n-1)^2*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 3, b(2) = 23 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 8 7 6 5 4 a(n) = 1/4 (91390 n - 815480 n + 3029921 n - 6092265 n + 7223517 n 3 2 / 2 - 5167735 n + 2187096 n - 504180 n + 48960) a(n - 1) / (n %1 / 2 (2 n - 1) ) + 2 (2 n - 3) 5 4 3 2 2 (8892 n - 57114 n + 134370 n - 140549 n + 62831 n - 10064) (n - 1) / 2 2 a(n - 2) / (n %1 (2 n - 1) ) + 16 (2 n - 3) (2 n - 5) / 4 3 2 2 / 2 (494 n - 1444 n + 1499 n - 653 n + 102) (n - 2) a(n - 3) / (n %1 / 2 (2 n - 1) ) 4 3 2 %1 := 494 n - 3420 n + 8795 n - 9959 n + 4192 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately -0.1875939585917285211446257548624099188936431880205669423991138362942556400\ -2140 041894338720446045067709965 10 This constant is identified as, 0 The implied delta is, -0.4570032917824090253740724696642378721653913713416813\ 085605005316561844432380597890427375868854996916 Since this is negative, there is no Apery-style irrationality proof of, 0, but still a very fast way to compute it to many digits ----------------------- This took, 1.311, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(2 n, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(2 n, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 8 7 6 5 4 b(n) = (429300 n - 3846528 n + 14331320 n - 28888111 n + 34362820 n 3 2 / 2 - 24699086 n + 10521279 n - 2445237 n + 239598) b(n - 1) / (n %1 / 2 7 6 5 4 (2 n - 1) ) - (2 n - 3) (42400 n - 358704 n + 1287904 n - 2490834 n 3 2 / 2 + 2757199 n - 1730634 n + 570699 n - 74358) b(n - 2) / (n %1 / 2 (2 n - 1) ) + 81 (2 n - 3) (2 n - 5) 4 3 2 2 / 2 (5300 n - 15688 n + 16192 n - 6947 n + 1071) (n - 2) b(n - 3) / (n / 2 %1 (2 n - 1) ) 4 3 2 %1 := 5300 n - 36888 n + 95056 n - 107595 n + 45198 and in Maple notation b(n) = (429300*n^8-3846528*n^7+14331320*n^6-28888111*n^5+34362820*n^4-24699086* n^3+10521279*n^2-2445237*n+239598)/n^2/(5300*n^4-36888*n^3+95056*n^2-107595*n+ 45198)/(2*n-1)^2*b(n-1)-(2*n-3)*(42400*n^7-358704*n^6+1287904*n^5-2490834*n^4+ 2757199*n^3-1730634*n^2+570699*n-74358)/n^2/(5300*n^4-36888*n^3+95056*n^2-\ 107595*n+45198)/(2*n-1)^2*b(n-2)+81*(2*n-3)*(2*n-5)*(5300*n^4-15688*n^3+16192*n ^2-6947*n+1071)*(n-2)^2/n^2/(5300*n^4-36888*n^3+95056*n^2-107595*n+45198)/(2*n-\ 1)^2*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 5, b(2) = 57 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 8 7 6 5 4 a(n) = (429300 n - 3846528 n + 14331320 n - 28888111 n + 34362820 n 3 2 / 2 - 24699086 n + 10521279 n - 2445237 n + 239598) a(n - 1) / (n %1 / 2 7 6 5 4 (2 n - 1) ) - (2 n - 3) (42400 n - 358704 n + 1287904 n - 2490834 n 3 2 / 2 + 2757199 n - 1730634 n + 570699 n - 74358) a(n - 2) / (n %1 / 2 (2 n - 1) ) + 81 (2 n - 3) (2 n - 5) 4 3 2 2 / 2 (5300 n - 15688 n + 16192 n - 6947 n + 1071) (n - 2) a(n - 3) / (n / 2 %1 (2 n - 1) ) 4 3 2 %1 := 5300 n - 36888 n + 95056 n - 107595 n + 45198 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.15403270679109895764827380476848368179455558541339005647126222433186524932\ 65988256901918504436485972 This constant is identified as, 2/9 ln(2) The implied delta is, -0.5365984093536425287473730021928375707968709447135228\ 575206690043799086168464469527846731603384502838 Since this is negative, there is no Apery-style irrationality proof of, 2/9 ln(2), but still a very fast way to compute it to many digits ----------------------- This took, 1.717, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(2 n, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(2 n, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 8 7 6 5 4 b(n) = 1/4 (959760 n - 8637840 n + 32299623 n - 65316510 n + 77940861 n 3 2 / 2 - 56212424 n + 24035422 n - 5609016 n + 552024) b(n - 1) / (n %1 / 2 7 6 5 4 (2 n - 1) ) - 4 (2 n - 3) (74304 n - 631584 n + 2200284 n - 4044870 n 3 2 / 2 + 4208280 n - 2463401 n + 750402 n - 92089) b(n - 2) / (n %1 / 2 (2 n - 1) ) + 256 (2 n - 3) (2 n - 5) 4 3 2 2 / 2 (2064 n - 6192 n + 6423 n - 2754 n + 425) (n - 2) b(n - 3) / (n %1 / 2 (2 n - 1) ) 4 3 2 %1 := 2064 n - 14448 n + 37383 n - 42432 n + 17858 and in Maple notation b(n) = 1/4*(959760*n^8-8637840*n^7+32299623*n^6-65316510*n^5+77940861*n^4-\ 56212424*n^3+24035422*n^2-5609016*n+552024)/n^2/(2064*n^4-14448*n^3+37383*n^2-\ 42432*n+17858)/(2*n-1)^2*b(n-1)-4*(2*n-3)*(74304*n^7-631584*n^6+2200284*n^5-\ 4044870*n^4+4208280*n^3-2463401*n^2+750402*n-92089)/n^2/(2064*n^4-14448*n^3+ 37383*n^2-42432*n+17858)/(2*n-1)^2*b(n-2)+256*(2*n-3)*(2*n-5)*(2064*n^4-6192*n^ 3+6423*n^2-2754*n+425)*(n-2)^2/n^2/(2064*n^4-14448*n^3+37383*n^2-42432*n+17858) /(2*n-1)^2*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 7, b(2) = 103 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 8 7 6 5 4 a(n) = 1/4 (959760 n - 8637840 n + 32299623 n - 65316510 n + 77940861 n 3 2 / 2 - 56212424 n + 24035422 n - 5609016 n + 552024) a(n - 1) / (n %1 / 2 7 6 5 4 (2 n - 1) ) - 4 (2 n - 3) (74304 n - 631584 n + 2200284 n - 4044870 n 3 2 / 2 + 4208280 n - 2463401 n + 750402 n - 92089) a(n - 2) / (n %1 / 2 (2 n - 1) ) + 256 (2 n - 3) (2 n - 5) 4 3 2 2 / 2 (2064 n - 6192 n + 6423 n - 2754 n + 425) (n - 2) a(n - 3) / (n %1 / 2 (2 n - 1) ) 4 3 2 %1 := 2064 n - 14448 n + 37383 n - 42432 n + 17858 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.12924850454918937545826414552029714172323418327326464138055227454558756390\ 80716431616018535074978928 This constant is identified as, 2/17 ln(3) The implied delta is, -0.5909381366514586888753273603886561207140128801021461\ 750056808644950711513441157751501173829712634520 Since this is negative, there is no Apery-style irrationality proof of, 2/17 ln(3), but still a very fast way to compute it to many digits ----------------------- This took, 1.751, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ ) binomial(n, k) binomial(n + k, k) binomial(2 n, k) / ----- k By Shalosh B. Ekhad Let ----- \ b(n) = ) binomial(n, k) binomial(n + k, k) binomial(2 n, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients b(n) = 1/4 6 5 4 3 2 (29665 n - 141345 n + 264772 n - 249181 n + 124975 n - 31902 n + 3276) / 2 2 2 b(n - 1) / (n (85 n - 235 n + 163) (2 n - 1) ) / 2 2 2 (85 n - 65 n + 13) (n - 1) (2 n - 3) b(n - 2) + ------------------------------------------------ 2 2 2 n (85 n - 235 n + 163) (2 n - 1) and in Maple notation b(n) = 1/4*(29665*n^6-141345*n^5+264772*n^4-249181*n^3+124975*n^2-31902*n+3276) /n^2/(85*n^2-235*n+163)/(2*n-1)^2*b(n-1)+(85*n^2-65*n+13)*(n-1)^2*(2*n-3)^2/n^2 /(85*n^2-235*n+163)/(2*n-1)^2*b(n-2) Of course, the initial conditions are b(0) = 1, b(1) = 5 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. a(n) = 1/4 6 5 4 3 2 (29665 n - 141345 n + 264772 n - 249181 n + 124975 n - 31902 n + 3276) / 2 2 2 a(n - 1) / (n (85 n - 235 n + 163) (2 n - 1) ) / 2 2 2 (85 n - 65 n + 13) (n - 1) (2 n - 3) a(n - 2) + ------------------------------------------------ 2 2 2 n (85 n - 235 n + 163) (2 n - 1) but with the following simpler initial conditions a(0) = 0, a(1) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.19938594749675471957241395959345759869320604863112708336188584598424332974\ 58425301616519879539101461 2 2 Pi This constant is identified as, ----- 99 The implied delta is, -0.1258541371681989545806168285292593001026961904084460\ 573649849848302616815941689599194050953982940120 2 2 Pi Since this is negative, there is no Apery-style irrationality proof of, -----, 99 but still a very fast way to compute it to many digits ----------------------- This took, 5.088, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + k, k) binomial(2 n, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + k, k) binomial(2 n, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 8 7 6 5 4 b(n) = (517924 n - 4490860 n + 16142592 n - 31341035 n + 35864044 n 3 2 / 2 - 24744828 n + 10070689 n - 2220117 n + 204750) b(n - 1) / (n %1 / 2 7 6 5 4 (2 n - 1) ) + (2 n - 3) (347600 n - 2840200 n + 9364704 n - 16005762 n 3 2 / 2 + 15130773 n - 7808334 n + 2025433 n - 207194) b(n - 2) / (n %1 / 2 (2 n - 1) ) + (2 n - 5) (2 n - 3) 4 3 2 2 / 2 (3476 n - 9284 n + 8148 n - 2847 n + 351) (n - 2) b(n - 3) / (n %1 / 2 (2 n - 1) ) 4 3 2 %1 := 3476 n - 23188 n + 56856 n - 60899 n + 24106 and in Maple notation b(n) = (517924*n^8-4490860*n^7+16142592*n^6-31341035*n^5+35864044*n^4-24744828* n^3+10070689*n^2-2220117*n+204750)/n^2/(3476*n^4-23188*n^3+56856*n^2-60899*n+ 24106)/(2*n-1)^2*b(n-1)+(2*n-3)*(347600*n^7-2840200*n^6+9364704*n^5-16005762*n^ 4+15130773*n^3-7808334*n^2+2025433*n-207194)/n^2/(3476*n^4-23188*n^3+56856*n^2-\ 60899*n+24106)/(2*n-1)^2*b(n-2)+(2*n-5)*(2*n-3)*(3476*n^4-9284*n^3+8148*n^2-\ 2847*n+351)*(n-2)^2/n^2/(3476*n^4-23188*n^3+56856*n^2-60899*n+24106)/(2*n-1)^2* b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 9, b(2) = 193 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 8 7 6 5 4 a(n) = (517924 n - 4490860 n + 16142592 n - 31341035 n + 35864044 n 3 2 / 2 - 24744828 n + 10070689 n - 2220117 n + 204750) a(n - 1) / (n %1 / 2 7 6 5 4 (2 n - 1) ) + (2 n - 3) (347600 n - 2840200 n + 9364704 n - 16005762 n 3 2 / 2 + 15130773 n - 7808334 n + 2025433 n - 207194) a(n - 2) / (n %1 / 2 (2 n - 1) ) + (2 n - 5) (2 n - 3) 4 3 2 2 / 2 (3476 n - 9284 n + 8148 n - 2847 n + 351) (n - 2) a(n - 3) / (n %1 / 2 (2 n - 1) ) 4 3 2 %1 := 3476 n - 23188 n + 56856 n - 60899 n + 24106 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.10663802777845312452572801868587331816546155913234696217241230915282978799\ 53376485547482041532951827 This constant is identified as, 2/13 ln(2) The implied delta is, -0.3952433249552788118673358318986997876400892276664200\ 779885995597374822737888469552259700099378448401 Since this is negative, there is no Apery-style irrationality proof of, 2/13 ln(2), but still a very fast way to compute it to many digits ----------------------- This took, 1.926, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + k, k) binomial(2 n, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + k, k) binomial(2 n, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 8 7 6 5 b(n) = 1/4 (6147099 n - 53004231 n + 189377076 n - 365393148 n 4 3 2 + 415600821 n - 285202285 n + 115586012 n - 25419000 n + 2342808) / 2 2 7 6 b(n - 1) / (n %1 (2 n - 1) ) + (2 n - 3) (4768092 n - 38729502 n / 5 4 3 2 + 126611520 n - 213747585 n + 198457095 n - 99729287 n + 24898929 n / 2 2 - 2423278) b(n - 2) / (n %1 (2 n - 1) ) + 4 (2 n - 5) (2 n - 3) / 4 3 2 2 / 2 (7791 n - 20433 n + 17388 n - 5806 n + 676) (n - 2) b(n - 3) / (n %1 / 2 (2 n - 1) ) 4 3 2 %1 := 7791 n - 51597 n + 125433 n - 133045 n + 52094 and in Maple notation b(n) = 1/4*(6147099*n^8-53004231*n^7+189377076*n^6-365393148*n^5+415600821*n^4-\ 285202285*n^3+115586012*n^2-25419000*n+2342808)/n^2/(7791*n^4-51597*n^3+125433* n^2-133045*n+52094)/(2*n-1)^2*b(n-1)+(2*n-3)*(4768092*n^7-38729502*n^6+ 126611520*n^5-213747585*n^4+198457095*n^3-99729287*n^2+24898929*n-2423278)/n^2/ (7791*n^4-51597*n^3+125433*n^2-133045*n+52094)/(2*n-1)^2*b(n-2)+4*(2*n-5)*(2*n-\ 3)*(7791*n^4-20433*n^3+17388*n^2-5806*n+676)*(n-2)^2/n^2/(7791*n^4-51597*n^3+ 125433*n^2-133045*n+52094)/(2*n-1)^2*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 13, b(2) = 397 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 8 7 6 5 a(n) = 1/4 (6147099 n - 53004231 n + 189377076 n - 365393148 n 4 3 2 + 415600821 n - 285202285 n + 115586012 n - 25419000 n + 2342808) / 2 2 7 6 a(n - 1) / (n %1 (2 n - 1) ) + (2 n - 3) (4768092 n - 38729502 n / 5 4 3 2 + 126611520 n - 213747585 n + 198457095 n - 99729287 n + 24898929 n / 2 2 - 2423278) a(n - 2) / (n %1 (2 n - 1) ) + 4 (2 n - 5) (2 n - 3) / 4 3 2 2 / 2 (7791 n - 20433 n + 17388 n - 5806 n + 676) (n - 2) a(n - 3) / (n %1 / 2 (2 n - 1) ) 4 3 2 %1 := 7791 n - 51597 n + 125433 n - 133045 n + 52094 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.06866326804175685571220282730765785654046815986392184073341839585234339332\ 616306042960098467585825555 This constant is identified as, 1/16 ln(3) The implied delta is, -0.6171763499291324152728837009681205660643491270280632\ 126361858320181632861695894207271015107929434481 Since this is negative, there is no Apery-style irrationality proof of, 1/16 ln(3), but still a very fast way to compute it to many digits ----------------------- This took, 1.338, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n, k) / ----- k By Shalosh B. Ekhad Let ----- \ b(n) = ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 16 15 14 b(n) = 1/12 (71491338804510 n - 1666398758256855 n + 17733671479303476 n 13 12 - 114242658666838228 n + 498024303580605992 n 11 10 - 1555654188021881918 n + 3596380511620656814 n 9 8 7 - 6266810657287698390 n + 8304801922105780422 n - 8383777746638443415 n 6 5 4 + 6415749765418823194 n - 3677530526675182354 n + 1546235775842192984 n 3 2 - 460761985033790904 n + 91798712800380864 n - 10941481772141760 n / 2 2 + 589606825248000) b(n - 1) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) + / 16 15 14 1/2 (33327126658692 n - 810152451362358 n + 9021808000975896 n 13 12 11 - 61023483364051689 n + 280237658895134256 n - 925004215217154042 n 10 9 + 2265884239085170355 n - 4192604375838509701 n 8 7 6 + 5906854625375559772 n - 6338784287323213530 n + 5146515481850746539 n 5 4 3 - 3116428663686158664 n + 1374409197919819474 n - 425277514480852544 n 2 + 86901970481679464 n - 10494359989960320 n + 568738523404800) b(n - 2) / 2 2 13 / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) - (2 n - 5) (201778768872 n / 12 11 10 - 3795277021632 n + 31677290925939 n - 154962033789942 n 9 8 7 + 494373298749519 n - 1082979961655036 n + 1670430486180844 n 6 5 4 - 1830061650137146 n + 1418223180977743 n - 764557024906372 n 3 2 + 277672357526707 n - 64251451737416 n + 8526011634072 n - 496946677752) 2 / 2 2 (n - 2) b(n - 3) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) + 9 (2 n - 5) / 10 9 8 (2 n - 7) (5604965802 n - 57782152395 n + 256095796368 n 7 6 5 4 - 641154450536 n + 1000788342896 n - 1013512384085 n + 671477047172 n 3 2 2 - 286402898264 n + 75249101426 n - 11066945328 n + 703318824) (n - 2) 2 / 2 2 (n - 3) b(n - 4) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) / 10 9 8 7 %1 := 5604965802 n - 113831810415 n + 1028358629013 n - 5442674203940 n 6 5 4 + 18690295414552 n - 43516853083199 n + 69575569645747 n 3 2 - 75431268848510 n + 53077575770410 n - 21891910563732 n + 4019837403096 and in Maple notation b(n) = 1/12*(71491338804510*n^16-1666398758256855*n^15+17733671479303476*n^14-\ 114242658666838228*n^13+498024303580605992*n^12-1555654188021881918*n^11+ 3596380511620656814*n^10-6266810657287698390*n^9+8304801922105780422*n^8-\ 8383777746638443415*n^7+6415749765418823194*n^6-3677530526675182354*n^5+ 1546235775842192984*n^4-460761985033790904*n^3+91798712800380864*n^2-\ 10941481772141760*n+589606825248000)/n^2/(3*n-1)/(3*n-2)/(5604965802*n^10-\ 113831810415*n^9+1028358629013*n^8-5442674203940*n^7+18690295414552*n^6-\ 43516853083199*n^5+69575569645747*n^4-75431268848510*n^3+53077575770410*n^2-\ 21891910563732*n+4019837403096)/(2*n-1)^2*b(n-1)+1/2*(33327126658692*n^16-\ 810152451362358*n^15+9021808000975896*n^14-61023483364051689*n^13+ 280237658895134256*n^12-925004215217154042*n^11+2265884239085170355*n^10-\ 4192604375838509701*n^9+5906854625375559772*n^8-6338784287323213530*n^7+ 5146515481850746539*n^6-3116428663686158664*n^5+1374409197919819474*n^4-\ 425277514480852544*n^3+86901970481679464*n^2-10494359989960320*n+ 568738523404800)/n^2/(3*n-1)/(3*n-2)/(5604965802*n^10-113831810415*n^9+ 1028358629013*n^8-5442674203940*n^7+18690295414552*n^6-43516853083199*n^5+ 69575569645747*n^4-75431268848510*n^3+53077575770410*n^2-21891910563732*n+ 4019837403096)/(2*n-1)^2*b(n-2)-(2*n-5)*(201778768872*n^13-3795277021632*n^12+ 31677290925939*n^11-154962033789942*n^10+494373298749519*n^9-1082979961655036*n ^8+1670430486180844*n^7-1830061650137146*n^6+1418223180977743*n^5-\ 764557024906372*n^4+277672357526707*n^3-64251451737416*n^2+8526011634072*n-\ 496946677752)*(n-2)^2/n^2/(3*n-1)/(3*n-2)/(5604965802*n^10-113831810415*n^9+ 1028358629013*n^8-5442674203940*n^7+18690295414552*n^6-43516853083199*n^5+ 69575569645747*n^4-75431268848510*n^3+53077575770410*n^2-21891910563732*n+ 4019837403096)/(2*n-1)^2*b(n-3)+9*(2*n-5)*(2*n-7)*(5604965802*n^10-57782152395* n^9+256095796368*n^8-641154450536*n^7+1000788342896*n^6-1013512384085*n^5+ 671477047172*n^4-286402898264*n^3+75249101426*n^2-11066945328*n+703318824)*(n-2 )^2*(n-3)^2/n^2/(3*n-1)/(3*n-2)/(5604965802*n^10-113831810415*n^9+1028358629013 *n^8-5442674203940*n^7+18690295414552*n^6-43516853083199*n^5+69575569645747*n^4 -75431268848510*n^3+53077575770410*n^2-21891910563732*n+4019837403096)/(2*n-1)^ 2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 7, b(2) = 123, b(3) = 2716 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 16 15 14 a(n) = 1/12 (71491338804510 n - 1666398758256855 n + 17733671479303476 n 13 12 - 114242658666838228 n + 498024303580605992 n 11 10 - 1555654188021881918 n + 3596380511620656814 n 9 8 7 - 6266810657287698390 n + 8304801922105780422 n - 8383777746638443415 n 6 5 4 + 6415749765418823194 n - 3677530526675182354 n + 1546235775842192984 n 3 2 - 460761985033790904 n + 91798712800380864 n - 10941481772141760 n / 2 2 + 589606825248000) a(n - 1) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) + / 16 15 14 1/2 (33327126658692 n - 810152451362358 n + 9021808000975896 n 13 12 11 - 61023483364051689 n + 280237658895134256 n - 925004215217154042 n 10 9 + 2265884239085170355 n - 4192604375838509701 n 8 7 6 + 5906854625375559772 n - 6338784287323213530 n + 5146515481850746539 n 5 4 3 - 3116428663686158664 n + 1374409197919819474 n - 425277514480852544 n 2 + 86901970481679464 n - 10494359989960320 n + 568738523404800) a(n - 2) / 2 2 13 / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) - (2 n - 5) (201778768872 n / 12 11 10 - 3795277021632 n + 31677290925939 n - 154962033789942 n 9 8 7 + 494373298749519 n - 1082979961655036 n + 1670430486180844 n 6 5 4 - 1830061650137146 n + 1418223180977743 n - 764557024906372 n 3 2 + 277672357526707 n - 64251451737416 n + 8526011634072 n - 496946677752) 2 / 2 2 (n - 2) a(n - 3) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) + 9 (2 n - 5) / 10 9 8 (2 n - 7) (5604965802 n - 57782152395 n + 256095796368 n 7 6 5 4 - 641154450536 n + 1000788342896 n - 1013512384085 n + 671477047172 n 3 2 2 - 286402898264 n + 75249101426 n - 11066945328 n + 703318824) (n - 2) 2 / 2 2 (n - 3) a(n - 4) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) / 10 9 8 7 %1 := 5604965802 n - 113831810415 n + 1028358629013 n - 5442674203940 n 6 5 4 + 18690295414552 n - 43516853083199 n + 69575569645747 n 3 2 - 75431268848510 n + 53077575770410 n - 21891910563732 n + 4019837403096 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately -0.1211691924130834951236694740442687531745740599126002343437847448859723297\ -2182 331172716026755910862306067 10 This constant is identified as, 0 The implied delta is, -0.5387894636305083296223914558830495625807614351949514\ 140637866548377784069234017930212018579533038314 Since this is negative, there is no Apery-style irrationality proof of, 0, but still a very fast way to compute it to many digits ----------------------- This took, 1.446, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 16 15 b(n) = 1/2 (1785513894476976 n - 41433592692573504 n 14 13 + 438853913247935232 n - 2813212811276760060 n 12 11 + 12201583588188398472 n - 37919090114350779980 n 10 9 + 87226649144719531488 n - 151296842342675336521 n 8 7 + 199716162597592307147 n - 201053624828789498518 n 6 5 + 153685756453599389450 n - 88199503362453504089 n 4 3 + 37240707837520654295 n - 11184080281473223008 n 2 + 2253886009712089980 n - 272495913427827360 n + 14900989869360000) / 2 2 b(n - 1) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) + 1/2 ( / 16 15 14 45017402710750344 n - 1089664959789078120 n + 12074958450717266628 n 13 12 - 81213230646565825266 n + 370512406796433183000 n 11 10 - 1213654960626755491613 n + 2946396878539259989885 n 9 8 - 5394293959327123456631 n + 7504611647054725611990 n 7 6 - 7932288774018612968214 n + 6323094981846071270322 n 5 4 - 3743796119859961014328 n + 1605933556921061441051 n 3 2 - 480145808477449353028 n + 94064468890789378620 n / 2 - 10806732717402776640 n + 555276148770672000) b(n - 2) / (n (3 n - 1) / 2 13 (3 n - 2) %1 (2 n - 1) ) - 1/2 (2 n - 5) (313623199029888 n 12 11 10 - 5866452404657856 n + 48655481661979632 n - 236278972563077772 n 9 8 7 + 747325617782646426 n - 1620320635569441514 n + 2468231539538925779 n 6 5 4 - 2662996907348923915 n + 2024932720705704167 n - 1066111461679150489 n 3 2 + 375911775607848656 n - 83856693919385574 n + 10654364566288812 n 2 / 2 - 593422099626240) (n - 2) b(n - 3) / (n (3 n - 1) (3 n - 2) %1 / 2 10 (2 n - 1) ) + 343/6 (2 n - 5) (2 n - 7) (593983331496 n 9 8 7 - 6061846994136 n + 26530768127364 n - 65389971454282 n 6 5 4 + 100090218303658 n - 98896920866911 n + 63509646711946 n 3 2 - 26037091592371 n + 6509042191456 n - 901933857420 n + 53869320000) 2 2 / 2 2 (n - 2) (n - 3) b(n - 4) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) / 10 9 8 %1 := 593983331496 n - 12001680309096 n + 107816640991908 n 7 6 5 - 567140608041610 n + 1934613173171408 n - 4471827167159293 n 4 3 2 + 7093679516292017 n - 7625694416501623 n + 5317007330276555 n - 2171568195482802 n + 394575292751040 and in Maple notation b(n) = 1/2*(1785513894476976*n^16-41433592692573504*n^15+438853913247935232*n^ 14-2813212811276760060*n^13+12201583588188398472*n^12-37919090114350779980*n^11 +87226649144719531488*n^10-151296842342675336521*n^9+199716162597592307147*n^8-\ 201053624828789498518*n^7+153685756453599389450*n^6-88199503362453504089*n^5+ 37240707837520654295*n^4-11184080281473223008*n^3+2253886009712089980*n^2-\ 272495913427827360*n+14900989869360000)/n^2/(3*n-1)/(3*n-2)/(593983331496*n^10-\ 12001680309096*n^9+107816640991908*n^8-567140608041610*n^7+1934613173171408*n^6 -4471827167159293*n^5+7093679516292017*n^4-7625694416501623*n^3+ 5317007330276555*n^2-2171568195482802*n+394575292751040)/(2*n-1)^2*b(n-1)+1/2*( 45017402710750344*n^16-1089664959789078120*n^15+12074958450717266628*n^14-\ 81213230646565825266*n^13+370512406796433183000*n^12-1213654960626755491613*n^ 11+2946396878539259989885*n^10-5394293959327123456631*n^9+ 7504611647054725611990*n^8-7932288774018612968214*n^7+6323094981846071270322*n^ 6-3743796119859961014328*n^5+1605933556921061441051*n^4-480145808477449353028*n ^3+94064468890789378620*n^2-10806732717402776640*n+555276148770672000)/n^2/(3*n -1)/(3*n-2)/(593983331496*n^10-12001680309096*n^9+107816640991908*n^8-\ 567140608041610*n^7+1934613173171408*n^6-4471827167159293*n^5+7093679516292017* n^4-7625694416501623*n^3+5317007330276555*n^2-2171568195482802*n+ 394575292751040)/(2*n-1)^2*b(n-2)-1/2*(2*n-5)*(313623199029888*n^13-\ 5866452404657856*n^12+48655481661979632*n^11-236278972563077772*n^10+ 747325617782646426*n^9-1620320635569441514*n^8+2468231539538925779*n^7-\ 2662996907348923915*n^6+2024932720705704167*n^5-1066111461679150489*n^4+ 375911775607848656*n^3-83856693919385574*n^2+10654364566288812*n-\ 593422099626240)*(n-2)^2/n^2/(3*n-1)/(3*n-2)/(593983331496*n^10-12001680309096* n^9+107816640991908*n^8-567140608041610*n^7+1934613173171408*n^6-\ 4471827167159293*n^5+7093679516292017*n^4-7625694416501623*n^3+5317007330276555 *n^2-2171568195482802*n+394575292751040)/(2*n-1)^2*b(n-3)+343/6*(2*n-5)*(2*n-7) *(593983331496*n^10-6061846994136*n^9+26530768127364*n^8-65389971454282*n^7+ 100090218303658*n^6-98896920866911*n^5+63509646711946*n^4-26037091592371*n^3+ 6509042191456*n^2-901933857420*n+53869320000)*(n-2)^2*(n-3)^2/n^2/(3*n-1)/(3*n-\ 2)/(593983331496*n^10-12001680309096*n^9+107816640991908*n^8-567140608041610*n^ 7+1934613173171408*n^6-4471827167159293*n^5+7093679516292017*n^4-\ 7625694416501623*n^3+5317007330276555*n^2-2171568195482802*n+394575292751040)/( 2*n-1)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 13, b(2) = 425, b(3) = 17401 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 16 15 a(n) = 1/2 (1785513894476976 n - 41433592692573504 n 14 13 + 438853913247935232 n - 2813212811276760060 n 12 11 + 12201583588188398472 n - 37919090114350779980 n 10 9 + 87226649144719531488 n - 151296842342675336521 n 8 7 + 199716162597592307147 n - 201053624828789498518 n 6 5 + 153685756453599389450 n - 88199503362453504089 n 4 3 + 37240707837520654295 n - 11184080281473223008 n 2 + 2253886009712089980 n - 272495913427827360 n + 14900989869360000) / 2 2 a(n - 1) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) + 1/2 ( / 16 15 14 45017402710750344 n - 1089664959789078120 n + 12074958450717266628 n 13 12 - 81213230646565825266 n + 370512406796433183000 n 11 10 - 1213654960626755491613 n + 2946396878539259989885 n 9 8 - 5394293959327123456631 n + 7504611647054725611990 n 7 6 - 7932288774018612968214 n + 6323094981846071270322 n 5 4 - 3743796119859961014328 n + 1605933556921061441051 n 3 2 - 480145808477449353028 n + 94064468890789378620 n / 2 - 10806732717402776640 n + 555276148770672000) a(n - 2) / (n (3 n - 1) / 2 13 (3 n - 2) %1 (2 n - 1) ) - 1/2 (2 n - 5) (313623199029888 n 12 11 10 - 5866452404657856 n + 48655481661979632 n - 236278972563077772 n 9 8 7 + 747325617782646426 n - 1620320635569441514 n + 2468231539538925779 n 6 5 4 - 2662996907348923915 n + 2024932720705704167 n - 1066111461679150489 n 3 2 + 375911775607848656 n - 83856693919385574 n + 10654364566288812 n 2 / 2 - 593422099626240) (n - 2) a(n - 3) / (n (3 n - 1) (3 n - 2) %1 / 2 10 (2 n - 1) ) + 343/6 (2 n - 5) (2 n - 7) (593983331496 n 9 8 7 - 6061846994136 n + 26530768127364 n - 65389971454282 n 6 5 4 + 100090218303658 n - 98896920866911 n + 63509646711946 n 3 2 - 26037091592371 n + 6509042191456 n - 901933857420 n + 53869320000) 2 2 / 2 2 (n - 2) (n - 3) a(n - 4) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) / 10 9 8 %1 := 593983331496 n - 12001680309096 n + 107816640991908 n 7 6 5 - 567140608041610 n + 1934613173171408 n - 4471827167159293 n 4 3 2 + 7093679516292017 n - 7625694416501623 n + 5317007330276555 n - 2171568195482802 n + 394575292751040 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.04780325383172036616670566354883976331555173340415553476694344893057887048\ 066860107626643634458059914 This constant is identified as, 2/29 ln(2) The implied delta is, -0.8001153806680037311784063279100046752675405310312556\ 781601345108896393629659834815999677067915159885 Since this is negative, there is no Apery-style irrationality proof of, 2/29 ln(2), but still a very fast way to compute it to many digits ----------------------- This took, 1.319, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 16 15 b(n) = 1/36 (3440596013277142122 n - 79641581326700220945 n 14 13 + 841490594252089447656 n - 5382241569653371502156 n 12 11 + 23301921238000471285828 n - 72338267837900201080318 n 10 9 + 166420250389128160999022 n - 289208405919656404502190 n 8 7 + 383485635135757388124102 n - 389208749369364366756565 n 6 5 + 301404002400081117144782 n - 176314029768923137582730 n 4 3 + 76426303510798336181512 n - 23737145158551717405048 n 2 + 4977645770060146862784 n - 627912666421118159040 n / 2 2 + 35710802707451270400) b(n - 1) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) / 16 15 ) + 1/6 (59175268776845093808 n - 1428941426336137989288 n 14 13 + 15790933048519512018582 n - 105866113109672709313441 n 12 11 + 481180023263983958525314 n - 1569236921816570162470244 n 10 9 + 3789808700108082813545675 n - 6895168573226787536558429 n 8 7 + 9520243716293846898714826 n - 9969624574405102717634810 n 6 5 + 7855539650756060370465593 n - 4583327399677915111789804 n 4 3 + 1929200051982706595339746 n - 562695352819937182980112 n 2 + 106707103209138208662344 n - 11756703635049646517760 n / 2 + 575125383831656140800) b(n - 2) / (n (3 n - 1) (3 n - 2) %1 / 2 13 (2 n - 1) ) - 1/3 (2 n - 5) (139324758020516484 n 12 11 - 2598074019947687112 n + 21467508676826505741 n 10 9 - 103775491836805587738 n + 326398842448508123199 n 8 7 - 702791655839154599876 n + 1061292148279194220834 n 6 5 - 1132460824865511151318 n + 848937553019285355871 n 4 3 - 438678803033019022468 n + 150867112811963954839 n 2 - 32545165522154817560 n + 3957146162230448808 n - 209303570148739704) 2 / 2 2 (n - 2) b(n - 3) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) + 1331/9 / 10 9 (2 n - 5) (2 n - 7) (67176836075466 n - 681684463246527 n 8 7 6 + 2962236507130380 n - 7235043050875940 n + 10946358262282208 n 5 4 3 - 10653200765574773 n + 6705118693140104 n - 2675288529923864 n 2 2 2 + 644547097664306 n - 85054613015232 n + 4798977647112) (n - 2) (n - 3) / 2 2 b(n - 4) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) / 10 9 8 %1 := 67176836075466 n - 1353452824001187 n + 12120354299745093 n 7 6 5 - 63534796113849872 n + 215902912306620556 n - 496973303867043875 n 4 3 2 + 784748936679941851 n - 839389402625599538 n + 582070631004573694 n - 236314764515390988 n + 42660507796575912 and in Maple notation b(n) = 1/36*(3440596013277142122*n^16-79641581326700220945*n^15+ 841490594252089447656*n^14-5382241569653371502156*n^13+23301921238000471285828* n^12-72338267837900201080318*n^11+166420250389128160999022*n^10-\ 289208405919656404502190*n^9+383485635135757388124102*n^8-\ 389208749369364366756565*n^7+301404002400081117144782*n^6-\ 176314029768923137582730*n^5+76426303510798336181512*n^4-\ 23737145158551717405048*n^3+4977645770060146862784*n^2-627912666421118159040*n+ 35710802707451270400)/n^2/(3*n-1)/(3*n-2)/(67176836075466*n^10-1353452824001187 *n^9+12120354299745093*n^8-63534796113849872*n^7+215902912306620556*n^6-\ 496973303867043875*n^5+784748936679941851*n^4-839389402625599538*n^3+ 582070631004573694*n^2-236314764515390988*n+42660507796575912)/(2*n-1)^2*b(n-1) +1/6*(59175268776845093808*n^16-1428941426336137989288*n^15+ 15790933048519512018582*n^14-105866113109672709313441*n^13+ 481180023263983958525314*n^12-1569236921816570162470244*n^11+ 3789808700108082813545675*n^10-6895168573226787536558429*n^9+ 9520243716293846898714826*n^8-9969624574405102717634810*n^7+ 7855539650756060370465593*n^6-4583327399677915111789804*n^5+ 1929200051982706595339746*n^4-562695352819937182980112*n^3+ 106707103209138208662344*n^2-11756703635049646517760*n+575125383831656140800)/n ^2/(3*n-1)/(3*n-2)/(67176836075466*n^10-1353452824001187*n^9+12120354299745093* n^8-63534796113849872*n^7+215902912306620556*n^6-496973303867043875*n^5+ 784748936679941851*n^4-839389402625599538*n^3+582070631004573694*n^2-\ 236314764515390988*n+42660507796575912)/(2*n-1)^2*b(n-2)-1/3*(2*n-5)*( 139324758020516484*n^13-2598074019947687112*n^12+21467508676826505741*n^11-\ 103775491836805587738*n^10+326398842448508123199*n^9-702791655839154599876*n^8+ 1061292148279194220834*n^7-1132460824865511151318*n^6+848937553019285355871*n^5 -438678803033019022468*n^4+150867112811963954839*n^3-32545165522154817560*n^2+ 3957146162230448808*n-209303570148739704)*(n-2)^2/n^2/(3*n-1)/(3*n-2)/( 67176836075466*n^10-1353452824001187*n^9+12120354299745093*n^8-\ 63534796113849872*n^7+215902912306620556*n^6-496973303867043875*n^5+ 784748936679941851*n^4-839389402625599538*n^3+582070631004573694*n^2-\ 236314764515390988*n+42660507796575912)/(2*n-1)^2*b(n-3)+1331/9*(2*n-5)*(2*n-7) *(67176836075466*n^10-681684463246527*n^9+2962236507130380*n^8-7235043050875940 *n^7+10946358262282208*n^6-10653200765574773*n^5+6705118693140104*n^4-\ 2675288529923864*n^3+644547097664306*n^2-85054613015232*n+4798977647112)*(n-2)^ 2*(n-3)^2/n^2/(3*n-1)/(3*n-2)/(67176836075466*n^10-1353452824001187*n^9+ 12120354299745093*n^8-63534796113849872*n^7+215902912306620556*n^6-\ 496973303867043875*n^5+784748936679941851*n^4-839389402625599538*n^3+ 582070631004573694*n^2-236314764515390988*n+42660507796575912)/(2*n-1)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 19, b(2) = 907, b(3) = 54136 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 16 15 a(n) = 1/36 (3440596013277142122 n - 79641581326700220945 n 14 13 + 841490594252089447656 n - 5382241569653371502156 n 12 11 + 23301921238000471285828 n - 72338267837900201080318 n 10 9 + 166420250389128160999022 n - 289208405919656404502190 n 8 7 + 383485635135757388124102 n - 389208749369364366756565 n 6 5 + 301404002400081117144782 n - 176314029768923137582730 n 4 3 + 76426303510798336181512 n - 23737145158551717405048 n 2 + 4977645770060146862784 n - 627912666421118159040 n / 2 2 + 35710802707451270400) a(n - 1) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) / 16 15 ) + 1/6 (59175268776845093808 n - 1428941426336137989288 n 14 13 + 15790933048519512018582 n - 105866113109672709313441 n 12 11 + 481180023263983958525314 n - 1569236921816570162470244 n 10 9 + 3789808700108082813545675 n - 6895168573226787536558429 n 8 7 + 9520243716293846898714826 n - 9969624574405102717634810 n 6 5 + 7855539650756060370465593 n - 4583327399677915111789804 n 4 3 + 1929200051982706595339746 n - 562695352819937182980112 n 2 + 106707103209138208662344 n - 11756703635049646517760 n / 2 + 575125383831656140800) a(n - 2) / (n (3 n - 1) (3 n - 2) %1 / 2 13 (2 n - 1) ) - 1/3 (2 n - 5) (139324758020516484 n 12 11 - 2598074019947687112 n + 21467508676826505741 n 10 9 - 103775491836805587738 n + 326398842448508123199 n 8 7 - 702791655839154599876 n + 1061292148279194220834 n 6 5 - 1132460824865511151318 n + 848937553019285355871 n 4 3 - 438678803033019022468 n + 150867112811963954839 n 2 - 32545165522154817560 n + 3957146162230448808 n - 209303570148739704) 2 / 2 2 (n - 2) a(n - 3) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) + 1331/9 / 10 9 (2 n - 5) (2 n - 7) (67176836075466 n - 681684463246527 n 8 7 6 + 2962236507130380 n - 7235043050875940 n + 10946358262282208 n 5 4 3 - 10653200765574773 n + 6705118693140104 n - 2675288529923864 n 2 2 2 + 644547097664306 n - 85054613015232 n + 4798977647112) (n - 2) (n - 3) / 2 2 a(n - 4) / (n (3 n - 1) (3 n - 2) %1 (2 n - 1) ) / 10 9 8 %1 := 67176836075466 n - 1353452824001187 n + 12120354299745093 n 7 6 5 - 63534796113849872 n + 215902912306620556 n - 496973303867043875 n 4 3 2 + 784748936679941851 n - 839389402625599538 n + 582070631004573694 n - 236314764515390988 n + 42660507796575912 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.02853538412124960237390247368629936895188287163175972601908296970486998164\ 204179134736664298217485945 This constant is identified as, 2/77 ln(3) The implied delta is, -0.9051242257588536981775154066983840261979792184970435\ 875644955098693670186036253332210795546429655589 Since this is negative, there is no Apery-style irrationality proof of, 2/77 ln(3), but still a very fast way to compute it to many digits ----------------------- This took, 1.955, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 ) binomial(n, k) binomial(2 n + k, k) / ----- k By Shalosh B. Ekhad Let ----- \ 2 b(n) = ) binomial(n, k) binomial(2 n + k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 2 4 (2 n - 1) b(n - 1) b(n) = --------------------- 2 n and in Maple notation b(n) = 4*(2*n-1)^2/n^2*b(n-1) The recurrence has order less than 2, so we can't do anything. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(2 n + k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(2 n + k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 7 6 5 4 3 b(n) = (140400 n - 1310400 n + 4948350 n - 9707606 n + 10586767 n 2 / 2 - 6385000 n + 1983217 n - 246880) b(n - 1) / ((2 n - 1) %1 n ) - ( / 8 7 6 5 4 975780 n - 11546730 n + 58262796 n - 163248857 n + 276707316 n 3 2 / 2 - 288896678 n + 179907497 n - 60286277 n + 8133280) b(n - 2) / (n / 4 3 2 (2 n - 1) (2 n - 3) %1) + (2340 n - 8970 n + 12188 n - 6801 n + 1264) 2 2 / 2 (n - 2) (2 n - 5) b(n - 3) / (n (2 n - 1) (2 n - 3) %1) / 4 3 2 %1 := 2340 n - 18330 n + 53138 n - 67447 n + 31563 and in Maple notation b(n) = (140400*n^7-1310400*n^6+4948350*n^5-9707606*n^4+10586767*n^3-6385000*n^2 +1983217*n-246880)/(2*n-1)/(2340*n^4-18330*n^3+53138*n^2-67447*n+31563)/n^2*b(n -1)-(975780*n^8-11546730*n^7+58262796*n^6-163248857*n^5+276707316*n^4-288896678 *n^3+179907497*n^2-60286277*n+8133280)/n^2/(2*n-1)/(2*n-3)/(2340*n^4-18330*n^3+ 53138*n^2-67447*n+31563)*b(n-2)+(2340*n^4-8970*n^3+12188*n^2-6801*n+1264)*(n-2) ^2*(2*n-5)^2/n^2/(2*n-1)/(2*n-3)/(2340*n^4-18330*n^3+53138*n^2-67447*n+31563)*b (n-3) Of course, the initial conditions are b(0) = 1, b(1) = 7, b(2) = 101 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 7 6 5 4 3 a(n) = (140400 n - 1310400 n + 4948350 n - 9707606 n + 10586767 n 2 / 2 - 6385000 n + 1983217 n - 246880) a(n - 1) / ((2 n - 1) %1 n ) - ( / 8 7 6 5 4 975780 n - 11546730 n + 58262796 n - 163248857 n + 276707316 n 3 2 / 2 - 288896678 n + 179907497 n - 60286277 n + 8133280) a(n - 2) / (n / 4 3 2 (2 n - 1) (2 n - 3) %1) + (2340 n - 8970 n + 12188 n - 6801 n + 1264) 2 2 / 2 (n - 2) (2 n - 5) a(n - 3) / (n (2 n - 1) (2 n - 3) %1) / 4 3 2 %1 := 2340 n - 18330 n + 53138 n - 67447 n + 31563 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.19804205158855580269063774898805044802157146696007292974876571699811246341\ 99127758873895219989767679 This constant is identified as, 2/7 ln(2) The implied delta is, -0.6430410792864752820422810168512460877923696825863681\ 555340811064424143864025293213811060738615229652 Since this is negative, there is no Apery-style irrationality proof of, 2/7 ln(2), but still a very fast way to compute it to many digits ----------------------- This took, 1.364, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(2 n + k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(2 n + k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 7 6 5 4 3 b(n) = 2 (165213 n - 1541988 n + 5822391 n - 11422678 n + 12462848 n 2 / 2 - 7525190 n + 2341724 n - 292160) b(n - 1) / ((2 n - 1) %1 n ) - 1/4 ( / 8 7 6 5 4 17717670 n - 209659095 n + 1057312692 n - 2959004165 n + 5006016516 n 3 2 / - 5212542530 n + 3234652664 n - 1079283128 n + 144958720) b(n - 2) / ( / 2 n (2 n - 1) (2 n - 3) %1) + 4 3 2 2 2 (3798 n - 14559 n + 19748 n - 10977 n + 2032) (n - 2) (2 n - 5) / 2 b(n - 3) / (n (2 n - 1) (2 n - 3) %1) / 4 3 2 %1 := 3798 n - 29751 n + 86213 n - 109342 n + 51114 and in Maple notation b(n) = 2*(165213*n^7-1541988*n^6+5822391*n^5-11422678*n^4+12462848*n^3-7525190* n^2+2341724*n-292160)/(2*n-1)/(3798*n^4-29751*n^3+86213*n^2-109342*n+51114)/n^2 *b(n-1)-1/4*(17717670*n^8-209659095*n^7+1057312692*n^6-2959004165*n^5+ 5006016516*n^4-5212542530*n^3+3234652664*n^2-1079283128*n+144958720)/n^2/(2*n-1 )/(2*n-3)/(3798*n^4-29751*n^3+86213*n^2-109342*n+51114)*b(n-2)+(3798*n^4-14559* n^3+19748*n^2-10977*n+2032)*(n-2)^2*(2*n-5)^2/n^2/(2*n-1)/(2*n-3)/(3798*n^4-\ 29751*n^3+86213*n^2-109342*n+51114)*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 10, b(2) = 196 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 7 6 5 4 3 a(n) = 2 (165213 n - 1541988 n + 5822391 n - 11422678 n + 12462848 n 2 / 2 - 7525190 n + 2341724 n - 292160) a(n - 1) / ((2 n - 1) %1 n ) - 1/4 ( / 8 7 6 5 4 17717670 n - 209659095 n + 1057312692 n - 2959004165 n + 5006016516 n 3 2 / - 5212542530 n + 3234652664 n - 1079283128 n + 144958720) a(n - 2) / ( / 2 n (2 n - 1) (2 n - 3) %1) + 4 3 2 2 2 (3798 n - 14559 n + 19748 n - 10977 n + 2032) (n - 2) (2 n - 5) / 2 a(n - 3) / (n (2 n - 1) (2 n - 3) %1) / 4 3 2 %1 := 3798 n - 29751 n + 86213 n - 109342 n + 51114 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.15694461266687281305646360527464652923535579397467849310495633337678489903\ 12298524105165364019617270 This constant is identified as, 1/7 ln(3) The implied delta is, -0.7387163603934049784327097940622849376547082179073877\ 090808059543106802428057882712630347919369304769 Since this is negative, there is no Apery-style irrationality proof of, 1/7 ln(3), but still a very fast way to compute it to many digits ----------------------- This took, 1.131, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ ) binomial(n, k) binomial(n + k, k) binomial(2 n + k, k) / ----- k By Shalosh B. Ekhad Let ----- \ b(n) = ) binomial(n, k) binomial(n + k, k) binomial(2 n + k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 7 6 5 4 3 b(n) = 1/4 (1550992 n - 14280832 n + 52658517 n - 99944475 n + 104650800 n 2 / 2 - 60062211 n + 17466825 n - 1987200) b(n - 1) / ((2 n - 1) %1 n ) + / 8 7 6 5 4 1/16 (1919978 n - 22478233 n + 111238909 n - 302384968 n + 490319339 n 3 2 / 2 - 480558727 n + 273743102 n - 81119980 n + 9411840) b(n - 2) / (n / (2 n - 1) (2 n - 3) %1) + 1/4 4 3 2 2 2 (6254 n - 23187 n + 28449 n - 12881 n + 1872) (n - 2) (2 n - 5) / 2 b(n - 3) / (n (2 n - 1) (2 n - 3) %1) / 4 3 2 %1 := 6254 n - 48203 n + 135534 n - 164356 n + 72643 and in Maple notation b(n) = 1/4*(1550992*n^7-14280832*n^6+52658517*n^5-99944475*n^4+104650800*n^3-\ 60062211*n^2+17466825*n-1987200)/(2*n-1)/(6254*n^4-48203*n^3+135534*n^2-164356* n+72643)/n^2*b(n-1)+1/16*(1919978*n^8-22478233*n^7+111238909*n^6-302384968*n^5+ 490319339*n^4-480558727*n^3+273743102*n^2-81119980*n+9411840)/n^2/(2*n-1)/(2*n-\ 3)/(6254*n^4-48203*n^3+135534*n^2-164356*n+72643)*b(n-2)+1/4*(6254*n^4-23187*n^ 3+28449*n^2-12881*n+1872)*(n-2)^2*(2*n-5)^2/n^2/(2*n-1)/(2*n-3)/(6254*n^4-48203 *n^3+135534*n^2-164356*n+72643)*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 7, b(2) = 121 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 7 6 5 4 3 a(n) = 1/4 (1550992 n - 14280832 n + 52658517 n - 99944475 n + 104650800 n 2 / 2 - 60062211 n + 17466825 n - 1987200) a(n - 1) / ((2 n - 1) %1 n ) + / 8 7 6 5 4 1/16 (1919978 n - 22478233 n + 111238909 n - 302384968 n + 490319339 n 3 2 / 2 - 480558727 n + 273743102 n - 81119980 n + 9411840) a(n - 2) / (n / (2 n - 1) (2 n - 3) %1) + 1/4 4 3 2 2 2 (6254 n - 23187 n + 28449 n - 12881 n + 1872) (n - 2) (2 n - 5) / 2 a(n - 3) / (n (2 n - 1) (2 n - 3) %1) / 4 3 2 %1 := 6254 n - 48203 n + 135534 n - 164356 n + 72643 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.14218403703793749936763735824783109088728207884312928289654974553710638399\ 37835314063309388710602436 This constant is identified as, 8/39 ln(2) The implied delta is, -0.1397644934796739777192589250530690451212418977148508\ 699852376546668571947837870642619350259297785470 Since this is negative, there is no Apery-style irrationality proof of, 8/39 ln(2), but still a very fast way to compute it to many digits ----------------------- This took, 1.446, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + k, k) binomial(2 n + k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + k, k) binomial(2 n + k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 7 6 5 4 b(n) = 1/9 (10995616 n - 102385408 n + 379899242 n - 722715416 n 3 2 / + 756851703 n - 434184126 n + 126255441 n - 14375880) b(n - 1) / ( / 2 8 7 6 (2 n - 1) %1 n ) + 1/9 (1878068 n - 22182754 n + 110463938 n 5 4 3 2 - 301261113 n + 488266368 n - 475854094 n + 267504185 n - 77411173 n / 2 + 8718840) b(n - 2) / (n (2 n - 1) (2 n - 3) %1) + 1/9 / 4 3 2 2 2 (10492 n - 39990 n + 49074 n - 21257 n + 2916) (n - 2) (2 n - 5) / 2 b(n - 3) / (n (2 n - 1) (2 n - 3) %1) / 4 3 2 %1 := 10492 n - 81958 n + 231996 n - 281343 n + 123729 and in Maple notation b(n) = 1/9*(10995616*n^7-102385408*n^6+379899242*n^5-722715416*n^4+756851703*n^ 3-434184126*n^2+126255441*n-14375880)/(2*n-1)/(10492*n^4-81958*n^3+231996*n^2-\ 281343*n+123729)/n^2*b(n-1)+1/9*(1878068*n^8-22182754*n^7+110463938*n^6-\ 301261113*n^5+488266368*n^4-475854094*n^3+267504185*n^2-77411173*n+8718840)/n^2 /(2*n-1)/(2*n-3)/(10492*n^4-81958*n^3+231996*n^2-281343*n+123729)*b(n-2)+1/9*( 10492*n^4-39990*n^3+49074*n^2-21257*n+2916)*(n-2)^2*(2*n-5)^2/n^2/(2*n-1)/(2*n-\ 3)/(10492*n^4-81958*n^3+231996*n^2-281343*n+123729)*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 13, b(2) = 421 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 7 6 5 4 a(n) = 1/9 (10995616 n - 102385408 n + 379899242 n - 722715416 n 3 2 / + 756851703 n - 434184126 n + 126255441 n - 14375880) a(n - 1) / ( / 2 8 7 6 (2 n - 1) %1 n ) + 1/9 (1878068 n - 22182754 n + 110463938 n 5 4 3 2 - 301261113 n + 488266368 n - 475854094 n + 267504185 n - 77411173 n / 2 + 8718840) a(n - 2) / (n (2 n - 1) (2 n - 3) %1) + 1/9 / 4 3 2 2 2 (10492 n - 39990 n + 49074 n - 21257 n + 2916) (n - 2) (2 n - 5) / 2 a(n - 3) / (n (2 n - 1) (2 n - 3) %1) / 4 3 2 %1 := 10492 n - 81958 n + 231996 n - 281343 n + 123729 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.07682496785207325132214985345640299429785081707710416375844481931151381139\ 453112129283730211275411813 18 18 This constant is identified as, -- ln(3) - -- ln(2) 95 95 The implied delta is, -0.1261062000996385863432395163014365287650243098493960\ 407059662857188203367434121078038391779419606337 Since this is negative, there is no Apery-style irrationality proof of, 18 18 -- ln(3) - -- ln(2), but still a very fast way to compute it to many digits 95 95 ----------------------- This took, 1.947, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + k, k) binomial(2 n + k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + k, k) binomial(2 n + k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 7 6 5 4 b(n) = 1/16 (31465980 n - 301703220 n + 1139319891 n - 2186144157 n 3 2 / + 2296428414 n - 1317951317 n + 383051609 n - 43591680) b(n - 1) / ( / 2 8 7 6 (2 n - 1) %1 n ) + 1/64 (13589154 n - 164268891 n + 832976811 n 5 4 3 2 - 2301585186 n + 3757274325 n - 3659327299 n + 2031351590 n / 2 - 569950684 n + 61332480) b(n - 2) / (n (2 n - 1) (2 n - 3) %1) + 1/16 / 4 3 2 2 2 (11526 n - 47121 n + 58659 n - 23669 n + 2880) (n - 2) (2 n - 5) / 2 b(n - 3) / (n (2 n - 1) (2 n - 3) %1) / 4 3 2 %1 := 11526 n - 93225 n + 269178 n - 328454 n + 143855 and in Maple notation b(n) = 1/16*(31465980*n^7-301703220*n^6+1139319891*n^5-2186144157*n^4+ 2296428414*n^3-1317951317*n^2+383051609*n-43591680)/(2*n-1)/(11526*n^4-93225*n^ 3+269178*n^2-328454*n+143855)/n^2*b(n-1)+1/64*(13589154*n^8-164268891*n^7+ 832976811*n^6-2301585186*n^5+3757274325*n^4-3659327299*n^3+2031351590*n^2-\ 569950684*n+61332480)/n^2/(2*n-1)/(2*n-3)/(11526*n^4-93225*n^3+269178*n^2-\ 328454*n+143855)*b(n-2)+1/16*(11526*n^4-47121*n^3+58659*n^2-23669*n+2880)*(n-2) ^2*(2*n-5)^2/n^2/(2*n-1)/(2*n-3)/(11526*n^4-93225*n^3+269178*n^2-328454*n+ 143855)*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 19, b(2) = 901 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 7 6 5 4 a(n) = 1/16 (31465980 n - 301703220 n + 1139319891 n - 2186144157 n 3 2 / + 2296428414 n - 1317951317 n + 383051609 n - 43591680) a(n - 1) / ( / 2 8 7 6 (2 n - 1) %1 n ) + 1/64 (13589154 n - 164268891 n + 832976811 n 5 4 3 2 - 2301585186 n + 3757274325 n - 3659327299 n + 2031351590 n / 2 - 569950684 n + 61332480) a(n - 2) / (n (2 n - 1) (2 n - 3) %1) + 1/16 / 4 3 2 2 2 (11526 n - 47121 n + 58659 n - 23669 n + 2880) (n - 2) (2 n - 5) / 2 a(n - 3) / (n (2 n - 1) (2 n - 3) %1) / 4 3 2 %1 := 11526 n - 93225 n + 269178 n - 328454 n + 143855 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.05260472181975422673174290395315701604635606142130487890407601103529928241\ 751414205039742156417925235 64 32 This constant is identified as, --- ln(2) - --- ln(3) 175 175 The implied delta is, -0.1233609432936533445508772225542104805516216781260622\ 662600559913689578199533266479555419346883068194 Since this is negative, there is no Apery-style irrationality proof of, 64 32 --- ln(2) - --- ln(3), 175 175 but still a very fast way to compute it to many digits ----------------------- This took, 1.755, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n + k, k) / ----- k By Shalosh B. Ekhad Let ----- \ b(n) = ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n + k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 12 11 10 b(n) = 1/10 (146609894904 n - 2976954986412 n + 26737231598396 n 9 8 7 - 140032754221253 n + 474720390285659 n - 1093280114042732 n 6 5 4 + 1746471943893718 n - 1940797725240509 n + 1481612029378463 n 3 2 - 753938165607294 n + 241627762274060 n - 43637328207000 n / 2 2 13 + 3355812180000) b(n - 1) / (%1 (2 n - 1) n ) - 1/20 (33879178168 n / 12 11 10 - 772624148824 n + 7925437171289 n - 48338075300090 n 9 8 7 + 195183073054951 n - 549627220011524 n + 1107012917787007 n 6 5 4 - 1608360572610690 n + 1677195411687081 n - 1231916888065592 n 3 2 + 614539448902384 n - 195272952864560 n + 35074647702000 n / 2 2 - 2674130760000) b(n - 2) / ((2 n - 3) %1 (2 n - 1) n ) + 1/80 ( / 12 11 10 74429256462 n - 1660165415535 n + 16494695633185 n 9 8 7 - 96248879340791 n + 366091311871193 n - 952244790127157 n 6 5 4 + 1728075255189639 n - 2190887968815701 n + 1911393380828861 n 3 2 - 1108948529100816 n + 402014168020660 n - 81015106176000 n 2 / 2 + 6841467360000) (n - 2) b(n - 3) / (n (2 n - 3) (2 n - 5) %1 / 2 8 7 6 (2 n - 1) ) - 1/20 (74953934 n - 772421287 n + 3311900009 n 5 4 3 2 - 7662423905 n + 10373271269 n - 8328085804 n + 3829612312 n 2 2 2 / 2 - 915516000 n + 87372000) (2 n - 7) (n - 2) (n - 3) b(n - 4) / (n / 2 (2 n - 3) (2 n - 5) %1 (2 n - 1) ) 8 7 6 5 %1 := 74953934 n - 1372052759 n + 10817559170 n - 47952091290 n 4 3 2 + 130645411354 n - 223915575459 n + 235675793702 n - 139242183172 n + 35355556520 and in Maple notation b(n) = 1/10*(146609894904*n^12-2976954986412*n^11+26737231598396*n^10-\ 140032754221253*n^9+474720390285659*n^8-1093280114042732*n^7+1746471943893718*n ^6-1940797725240509*n^5+1481612029378463*n^4-753938165607294*n^3+ 241627762274060*n^2-43637328207000*n+3355812180000)/(74953934*n^8-1372052759*n^ 7+10817559170*n^6-47952091290*n^5+130645411354*n^4-223915575459*n^3+ 235675793702*n^2-139242183172*n+35355556520)/(2*n-1)^2/n^2*b(n-1)-1/20*( 33879178168*n^13-772624148824*n^12+7925437171289*n^11-48338075300090*n^10+ 195183073054951*n^9-549627220011524*n^8+1107012917787007*n^7-1608360572610690*n ^6+1677195411687081*n^5-1231916888065592*n^4+614539448902384*n^3-\ 195272952864560*n^2+35074647702000*n-2674130760000)/(2*n-3)/(74953934*n^8-\ 1372052759*n^7+10817559170*n^6-47952091290*n^5+130645411354*n^4-223915575459*n^ 3+235675793702*n^2-139242183172*n+35355556520)/(2*n-1)^2/n^2*b(n-2)+1/80*( 74429256462*n^12-1660165415535*n^11+16494695633185*n^10-96248879340791*n^9+ 366091311871193*n^8-952244790127157*n^7+1728075255189639*n^6-2190887968815701*n ^5+1911393380828861*n^4-1108948529100816*n^3+402014168020660*n^2-81015106176000 *n+6841467360000)*(n-2)^2/n^2/(2*n-3)/(2*n-5)/(74953934*n^8-1372052759*n^7+ 10817559170*n^6-47952091290*n^5+130645411354*n^4-223915575459*n^3+235675793702* n^2-139242183172*n+35355556520)/(2*n-1)^2*b(n-3)-1/20*(74953934*n^8-772421287*n ^7+3311900009*n^6-7662423905*n^5+10373271269*n^4-8328085804*n^3+3829612312*n^2-\ 915516000*n+87372000)*(2*n-7)^2*(n-2)^2*(n-3)^2/n^2/(2*n-3)/(2*n-5)/(74953934*n ^8-1372052759*n^7+10817559170*n^6-47952091290*n^5+130645411354*n^4-223915575459 *n^3+235675793702*n^2-139242183172*n+35355556520)/(2*n-1)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 10, b(2) = 266, b(3) = 8926 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 12 11 10 a(n) = 1/10 (146609894904 n - 2976954986412 n + 26737231598396 n 9 8 7 - 140032754221253 n + 474720390285659 n - 1093280114042732 n 6 5 4 + 1746471943893718 n - 1940797725240509 n + 1481612029378463 n 3 2 - 753938165607294 n + 241627762274060 n - 43637328207000 n / 2 2 13 + 3355812180000) a(n - 1) / (%1 (2 n - 1) n ) - 1/20 (33879178168 n / 12 11 10 - 772624148824 n + 7925437171289 n - 48338075300090 n 9 8 7 + 195183073054951 n - 549627220011524 n + 1107012917787007 n 6 5 4 - 1608360572610690 n + 1677195411687081 n - 1231916888065592 n 3 2 + 614539448902384 n - 195272952864560 n + 35074647702000 n / 2 2 - 2674130760000) a(n - 2) / ((2 n - 3) %1 (2 n - 1) n ) + 1/80 ( / 12 11 10 74429256462 n - 1660165415535 n + 16494695633185 n 9 8 7 - 96248879340791 n + 366091311871193 n - 952244790127157 n 6 5 4 + 1728075255189639 n - 2190887968815701 n + 1911393380828861 n 3 2 - 1108948529100816 n + 402014168020660 n - 81015106176000 n 2 / 2 + 6841467360000) (n - 2) a(n - 3) / (n (2 n - 3) (2 n - 5) %1 / 2 8 7 6 (2 n - 1) ) - 1/20 (74953934 n - 772421287 n + 3311900009 n 5 4 3 2 - 7662423905 n + 10373271269 n - 8328085804 n + 3829612312 n 2 2 2 / 2 - 915516000 n + 87372000) (2 n - 7) (n - 2) (n - 3) a(n - 4) / (n / 2 (2 n - 3) (2 n - 5) %1 (2 n - 1) ) 8 7 6 5 %1 := 74953934 n - 1372052759 n + 10817559170 n - 47952091290 n 4 3 2 + 130645411354 n - 223915575459 n + 235675793702 n - 139242183172 n + 35355556520 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.10009510254976838113707749603502451754059783314890457087372234976571191057\ 23562700840675034179581292 The implied delta is, -0.2000392997775032139315850967256430953307581487591209\ 029643385253616599571310639785806791087354222165 Since this is negative, there is no Apery-style irrationality proof of, 0.100\ 095102549768381137077496035024517540597833148904570873722349765711910572\ 3562700840675034179581292, but still a very fast way to compute it to many digits ----------------------- This took, 4.081, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n + k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n + k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 12 11 10 b(n) = 1/72 (47834432977604 n - 972161612084754 n + 8739541335993072 n 9 8 7 - 45816627284626437 n + 155477386044656910 n - 358437401719164468 n 6 5 4 + 573217795495357144 n - 637752815478902313 n + 487508449682062278 n 3 2 - 248458816223792964 n + 79776426498352920 n - 14440283708980032 n / 2 2 + 1113517795057920) b(n - 1) / (%1 (2 n - 1) n ) - 1/72 ( / 13 12 11 2433123423904 n - 55532316852064 n + 570098492417310 n 10 9 8 - 3479771806465888 n + 14060664861898489 n - 39617097156660195 n 7 6 5 + 79826057506349357 n - 115998726859960721 n + 120947049654877752 n 4 3 2 - 88786203982344152 n + 44237933676423152 n - 14027167936588512 n / + 2510880103460448 n - 190442160478080) b(n - 2) / ((2 n - 3) %1 / 2 2 12 11 (2 n - 1) n ) + 1/72 (1492573545444 n - 33319419157482 n 10 9 8 + 331355554591778 n - 1935516719252353 n + 7370314706130691 n 7 6 5 - 19194660438581279 n + 34879030334876266 n - 44281303042907300 n 4 3 2 + 38687455529113202 n - 22478378940225658 n + 8160846501620175 n 2 / 2 - 1647037436844924 n + 139291398595440) (n - 2) b(n - 3) / (n / 2 8 7 (2 n - 5) (2 n - 3) %1 (2 n - 1) ) - 1/72 (1758037156 n - 18149041946 n 6 5 4 3 + 77973078562 n - 180796907885 n + 245328717475 n - 197406886169 n 2 2 2 + 90954584787 n - 21774620556 n + 2079866160) (2 n - 7) (n - 2) 2 / 2 2 (n - 3) b(n - 4) / (n (2 n - 5) (2 n - 3) %1 (2 n - 1) ) / 8 7 6 5 %1 := 1758037156 n - 32213339194 n + 254241412552 n - 1128215340859 n 4 3 2 + 3077188504360 n - 5279818955005 n + 5563067726658 n - 3290149920204 n + 836221740696 and in Maple notation b(n) = 1/72*(47834432977604*n^12-972161612084754*n^11+8739541335993072*n^10-\ 45816627284626437*n^9+155477386044656910*n^8-358437401719164468*n^7+ 573217795495357144*n^6-637752815478902313*n^5+487508449682062278*n^4-\ 248458816223792964*n^3+79776426498352920*n^2-14440283708980032*n+ 1113517795057920)/(1758037156*n^8-32213339194*n^7+254241412552*n^6-\ 1128215340859*n^5+3077188504360*n^4-5279818955005*n^3+5563067726658*n^2-\ 3290149920204*n+836221740696)/(2*n-1)^2/n^2*b(n-1)-1/72*(2433123423904*n^13-\ 55532316852064*n^12+570098492417310*n^11-3479771806465888*n^10+ 14060664861898489*n^9-39617097156660195*n^8+79826057506349357*n^7-\ 115998726859960721*n^6+120947049654877752*n^5-88786203982344152*n^4+ 44237933676423152*n^3-14027167936588512*n^2+2510880103460448*n-190442160478080) /(2*n-3)/(1758037156*n^8-32213339194*n^7+254241412552*n^6-1128215340859*n^5+ 3077188504360*n^4-5279818955005*n^3+5563067726658*n^2-3290149920204*n+ 836221740696)/(2*n-1)^2/n^2*b(n-2)+1/72*(1492573545444*n^12-33319419157482*n^11 +331355554591778*n^10-1935516719252353*n^9+7370314706130691*n^8-\ 19194660438581279*n^7+34879030334876266*n^6-44281303042907300*n^5+ 38687455529113202*n^4-22478378940225658*n^3+8160846501620175*n^2-\ 1647037436844924*n+139291398595440)*(n-2)^2/n^2/(2*n-5)/(2*n-3)/(1758037156*n^8 -32213339194*n^7+254241412552*n^6-1128215340859*n^5+3077188504360*n^4-\ 5279818955005*n^3+5563067726658*n^2-3290149920204*n+836221740696)/(2*n-1)^2*b(n -3)-1/72*(1758037156*n^8-18149041946*n^7+77973078562*n^6-180796907885*n^5+ 245328717475*n^4-197406886169*n^3+90954584787*n^2-21774620556*n+2079866160)*(2* n-7)^2*(n-2)^2*(n-3)^2/n^2/(2*n-5)/(2*n-3)/(1758037156*n^8-32213339194*n^7+ 254241412552*n^6-1128215340859*n^5+3077188504360*n^4-5279818955005*n^3+ 5563067726658*n^2-3290149920204*n+836221740696)/(2*n-1)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 19, b(2) = 981, b(3) = 63715 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 12 11 10 a(n) = 1/72 (47834432977604 n - 972161612084754 n + 8739541335993072 n 9 8 7 - 45816627284626437 n + 155477386044656910 n - 358437401719164468 n 6 5 4 + 573217795495357144 n - 637752815478902313 n + 487508449682062278 n 3 2 - 248458816223792964 n + 79776426498352920 n - 14440283708980032 n / 2 2 + 1113517795057920) a(n - 1) / (%1 (2 n - 1) n ) - 1/72 ( / 13 12 11 2433123423904 n - 55532316852064 n + 570098492417310 n 10 9 8 - 3479771806465888 n + 14060664861898489 n - 39617097156660195 n 7 6 5 + 79826057506349357 n - 115998726859960721 n + 120947049654877752 n 4 3 2 - 88786203982344152 n + 44237933676423152 n - 14027167936588512 n / + 2510880103460448 n - 190442160478080) a(n - 2) / ((2 n - 3) %1 / 2 2 12 11 (2 n - 1) n ) + 1/72 (1492573545444 n - 33319419157482 n 10 9 8 + 331355554591778 n - 1935516719252353 n + 7370314706130691 n 7 6 5 - 19194660438581279 n + 34879030334876266 n - 44281303042907300 n 4 3 2 + 38687455529113202 n - 22478378940225658 n + 8160846501620175 n 2 / 2 - 1647037436844924 n + 139291398595440) (n - 2) a(n - 3) / (n / 2 8 7 (2 n - 5) (2 n - 3) %1 (2 n - 1) ) - 1/72 (1758037156 n - 18149041946 n 6 5 4 3 + 77973078562 n - 180796907885 n + 245328717475 n - 197406886169 n 2 2 2 + 90954584787 n - 21774620556 n + 2079866160) (2 n - 7) (n - 2) 2 / 2 2 (n - 3) a(n - 4) / (n (2 n - 5) (2 n - 3) %1 (2 n - 1) ) / 8 7 6 5 %1 := 1758037156 n - 32213339194 n + 254241412552 n - 1128215340859 n 4 3 2 + 3077188504360 n - 5279818955005 n + 5563067726658 n - 3290149920204 n + 836221740696 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.05264408966278065641143535099682353681586076969824723449017822856911850293\ 440719359031873369593053323 This constant is identified as, 6/79 ln(2) The implied delta is, -0.1997716735644068679762539748894298709925409479812602\ 288235781049042999046797502816990097658494464349 Since this is negative, there is no Apery-style irrationality proof of, 6/79 ln(2), but still a very fast way to compute it to many digits ----------------------- This took, 2.207, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n + k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n + k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 12 11 b(n) = 1/78 (520640472222444 n - 10583457095988714 n 10 9 8 + 95164505402148540 n - 499013534581291009 n + 1693813164797626561 n 7 6 5 - 3905966433389012268 n + 6248314802292447228 n - 6954092579437103727 n 4 3 2 + 5317904651700860943 n - 2711568621226108346 n + 871165703511378020 n / 2 2 - 157808001427639992 n + 12180063967737120) b(n - 1) / (%1 (2 n - 1) n / 13 12 11 ) - 1/156 (33387526964052 n - 762162614795592 n + 7825823592520497 n 10 9 8 - 47774676720609192 n + 193063365520685811 n - 543995269053870884 n 7 6 5 + 1096067332392412011 n - 1592485357874750244 n + 1659894514265326869 n 4 3 2 - 1217876229998049672 n + 606314197588649928 n - 192014520553773136 n / + 34307195121577872 n - 2595172672048320) b(n - 2) / ((2 n - 3) %1 / 2 2 12 11 (2 n - 1) n ) + 1/624 (86471550027882 n - 1930716436110831 n 10 9 8 + 19204998656010513 n - 112209823311119685 n + 427413544695521739 n 7 6 5 - 1113485528747432189 n + 2024058650501395355 n - 2570663152639215083 n 4 3 2 + 2246835980698290611 n - 1306026061642028372 n + 474369242184320300 n 2 / 2 - 95783134562378640 n + 8104424158958400) (n - 2) b(n - 3) / (n / 2 8 (2 n - 3) (2 n - 5) %1 (2 n - 1) ) - 1/156 (11915605626 n 7 6 5 4 - 123061573071 n + 528956544033 n - 1227146075603 n + 1666090416651 n 3 2 - 1341394923786 n + 618359513870 n - 148098174120 n + 14150635200) 2 2 2 / 2 (2 n - 7) (n - 2) (n - 3) b(n - 4) / (n (2 n - 3) (2 n - 5) %1 / 2 (2 n - 1) ) 8 7 6 5 %1 := 11915605626 n - 218386418079 n + 1724024513058 n - 7652452289348 n 4 3 2 + 20877416406466 n - 35830777199621 n + 37762825693678 n - 22339589138540 n + 5679173461960 and in Maple notation b(n) = 1/78*(520640472222444*n^12-10583457095988714*n^11+95164505402148540*n^10 -499013534581291009*n^9+1693813164797626561*n^8-3905966433389012268*n^7+ 6248314802292447228*n^6-6954092579437103727*n^5+5317904651700860943*n^4-\ 2711568621226108346*n^3+871165703511378020*n^2-157808001427639992*n+ 12180063967737120)/(11915605626*n^8-218386418079*n^7+1724024513058*n^6-\ 7652452289348*n^5+20877416406466*n^4-35830777199621*n^3+37762825693678*n^2-\ 22339589138540*n+5679173461960)/(2*n-1)^2/n^2*b(n-1)-1/156*(33387526964052*n^13 -762162614795592*n^12+7825823592520497*n^11-47774676720609192*n^10+ 193063365520685811*n^9-543995269053870884*n^8+1096067332392412011*n^7-\ 1592485357874750244*n^6+1659894514265326869*n^5-1217876229998049672*n^4+ 606314197588649928*n^3-192014520553773136*n^2+34307195121577872*n-\ 2595172672048320)/(2*n-3)/(11915605626*n^8-218386418079*n^7+1724024513058*n^6-\ 7652452289348*n^5+20877416406466*n^4-35830777199621*n^3+37762825693678*n^2-\ 22339589138540*n+5679173461960)/(2*n-1)^2/n^2*b(n-2)+1/624*(86471550027882*n^12 -1930716436110831*n^11+19204998656010513*n^10-112209823311119685*n^9+ 427413544695521739*n^8-1113485528747432189*n^7+2024058650501395355*n^6-\ 2570663152639215083*n^5+2246835980698290611*n^4-1306026061642028372*n^3+ 474369242184320300*n^2-95783134562378640*n+8104424158958400)*(n-2)^2/n^2/(2*n-3 )/(2*n-5)/(11915605626*n^8-218386418079*n^7+1724024513058*n^6-7652452289348*n^5 +20877416406466*n^4-35830777199621*n^3+37762825693678*n^2-22339589138540*n+ 5679173461960)/(2*n-1)^2*b(n-3)-1/156*(11915605626*n^8-123061573071*n^7+ 528956544033*n^6-1227146075603*n^5+1666090416651*n^4-1341394923786*n^3+ 618359513870*n^2-148098174120*n+14150635200)*(2*n-7)^2*(n-2)^2*(n-3)^2/n^2/(2*n -3)/(2*n-5)/(11915605626*n^8-218386418079*n^7+1724024513058*n^6-7652452289348*n ^5+20877416406466*n^4-35830777199621*n^3+37762825693678*n^2-22339589138540*n+ 5679173461960)/(2*n-1)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 28, b(2) = 2146, b(3) = 206704 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 12 11 a(n) = 1/78 (520640472222444 n - 10583457095988714 n 10 9 8 + 95164505402148540 n - 499013534581291009 n + 1693813164797626561 n 7 6 5 - 3905966433389012268 n + 6248314802292447228 n - 6954092579437103727 n 4 3 2 + 5317904651700860943 n - 2711568621226108346 n + 871165703511378020 n / 2 2 - 157808001427639992 n + 12180063967737120) a(n - 1) / (%1 (2 n - 1) n / 13 12 11 ) - 1/156 (33387526964052 n - 762162614795592 n + 7825823592520497 n 10 9 8 - 47774676720609192 n + 193063365520685811 n - 543995269053870884 n 7 6 5 + 1096067332392412011 n - 1592485357874750244 n + 1659894514265326869 n 4 3 2 - 1217876229998049672 n + 606314197588649928 n - 192014520553773136 n / + 34307195121577872 n - 2595172672048320) a(n - 2) / ((2 n - 3) %1 / 2 2 12 11 (2 n - 1) n ) + 1/624 (86471550027882 n - 1930716436110831 n 10 9 8 + 19204998656010513 n - 112209823311119685 n + 427413544695521739 n 7 6 5 - 1113485528747432189 n + 2024058650501395355 n - 2570663152639215083 n 4 3 2 + 2246835980698290611 n - 1306026061642028372 n + 474369242184320300 n 2 / 2 - 95783134562378640 n + 8104424158958400) (n - 2) a(n - 3) / (n / 2 8 (2 n - 3) (2 n - 5) %1 (2 n - 1) ) - 1/156 (11915605626 n 7 6 5 4 - 123061573071 n + 528956544033 n - 1227146075603 n + 1666090416651 n 3 2 - 1341394923786 n + 618359513870 n - 148098174120 n + 14150635200) 2 2 2 / 2 (2 n - 7) (n - 2) (n - 3) a(n - 4) / (n (2 n - 3) (2 n - 5) %1 / 2 (2 n - 1) ) 8 7 6 5 %1 := 11915605626 n - 218386418079 n + 1724024513058 n - 7652452289348 n 4 3 2 + 20877416406466 n - 35830777199621 n + 37762825693678 n - 22339589138540 n + 5679173461960 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.03571803942808124016883688626862374001327701462411091014323961646474918089\ 830332313779471632847152211 The implied delta is, -0.2012224203519101370334405816119339573371475647225492\ 362646421198527160382688645591295778640910171385 Since this is negative, there is no Apery-style irrationality proof of, 0.035\ 718039428081240168836886268623740013277014624110910143239616464749180898\ 30332313779471632847152211, but still a very fast way to compute it to many digits ----------------------- This took, 4.698, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 ) binomial(n, k) binomial(2 n + 2 k, k) / ----- k By Shalosh B. Ekhad Let ----- \ 2 b(n) = ) binomial(n, k) binomial(2 n + 2 k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 17 16 15 b(n) = 2/9 (23483364611520 n - 633248873215200 n + 7842268620985488 n 14 13 12 - 59176259116732588 n + 304375269420925548 n - 1131055969648436264 n 11 10 + 3139945515837906770 n - 6641753669827708749 n 9 8 + 10820005543539333064 n - 13630552859799932252 n 7 6 + 13256044178949879666 n - 9879674713109900653 n 5 4 3 + 5565183894263052808 n - 2316395259494917614 n + 687169983663160736 n 2 - 136767051939113880 n + 16293519136641600 n - 874779810912000) b(n - 1) / 2 2 2 17 / (n %1 (3 n - 1) (3 n - 2) ) - 4/9 (27817938441120 n / 16 15 14 - 777952276702320 n + 10030774054856688 n - 79141267090472332 n 13 12 + 427587074052228988 n - 1677251569760774892 n 11 10 + 4940837921091813374 n - 11150312161917911393 n 9 8 + 19487851717711047966 n - 26482263332780424296 n 7 6 + 27924525365091408845 n - 22665508662551929176 n 5 4 + 13949866511526698263 n - 6354066664650926851 n 3 2 + 2061398549040087156 n - 447186995298663540 n + 57710281016102400 n / 2 2 2 - 3327420122208000) b(n - 2) / (n %1 (3 n - 1) (3 n - 2) ) - 4/9 ( / 15 14 13 3031252991040 n - 75677805628320 n + 858344525581776 n 12 11 10 - 5857769424107608 n + 26854371245246884 n - 87440281467938624 n 9 8 7 + 208486612373084912 n - 369859713845792208 n + 491028173462289951 n 6 5 4 - 486556397579760371 n + 355858012913663087 n - 188041153939272139 n 3 2 + 69248122603406110 n - 16717449784653690 n + 2358040920379200 n 2 / 2 2 2 - 146268798354000) (n - 2) b(n - 3) / (n %1 (3 n - 1) (3 n - 2) ) - 4 / 11 10 9 8 (2948689680 n - 38232266280 n + 217211366892 n - 712740293240 n 7 6 5 + 1498419303272 n - 2115073845538 n + 2040519733965 n 4 3 2 - 1341682903844 n + 587361445501 n - 162521972848 n + 25510883640 n 2 2 2 / 2 2 - 1720270800) (2 n - 7) (n - 2) (n - 3) b(n - 4) / (n %1 (3 n - 1) / 2 (3 n - 2) ) 11 10 9 8 %1 := 2948689680 n - 70667852760 n + 761711962092 n - 4874628375068 n 7 6 5 + 20580890405304 n - 60197562549050 n + 124476682560457 n 4 3 2 - 181977361139651 n + 184339059916975 n - 123231781457679 n + 48932930544400 n - 8743942975500 and in Maple notation b(n) = 2/9*(23483364611520*n^17-633248873215200*n^16+7842268620985488*n^15-\ 59176259116732588*n^14+304375269420925548*n^13-1131055969648436264*n^12+ 3139945515837906770*n^11-6641753669827708749*n^10+10820005543539333064*n^9-\ 13630552859799932252*n^8+13256044178949879666*n^7-9879674713109900653*n^6+ 5565183894263052808*n^5-2316395259494917614*n^4+687169983663160736*n^3-\ 136767051939113880*n^2+16293519136641600*n-874779810912000)/n^2/(2948689680*n^ 11-70667852760*n^10+761711962092*n^9-4874628375068*n^8+20580890405304*n^7-\ 60197562549050*n^6+124476682560457*n^5-181977361139651*n^4+184339059916975*n^3-\ 123231781457679*n^2+48932930544400*n-8743942975500)/(3*n-1)^2/(3*n-2)^2*b(n-1)-\ 4/9*(27817938441120*n^17-777952276702320*n^16+10030774054856688*n^15-\ 79141267090472332*n^14+427587074052228988*n^13-1677251569760774892*n^12+ 4940837921091813374*n^11-11150312161917911393*n^10+19487851717711047966*n^9-\ 26482263332780424296*n^8+27924525365091408845*n^7-22665508662551929176*n^6+ 13949866511526698263*n^5-6354066664650926851*n^4+2061398549040087156*n^3-\ 447186995298663540*n^2+57710281016102400*n-3327420122208000)/n^2/(2948689680*n^ 11-70667852760*n^10+761711962092*n^9-4874628375068*n^8+20580890405304*n^7-\ 60197562549050*n^6+124476682560457*n^5-181977361139651*n^4+184339059916975*n^3-\ 123231781457679*n^2+48932930544400*n-8743942975500)/(3*n-1)^2/(3*n-2)^2*b(n-2)-\ 4/9*(3031252991040*n^15-75677805628320*n^14+858344525581776*n^13-\ 5857769424107608*n^12+26854371245246884*n^11-87440281467938624*n^10+ 208486612373084912*n^9-369859713845792208*n^8+491028173462289951*n^7-\ 486556397579760371*n^6+355858012913663087*n^5-188041153939272139*n^4+ 69248122603406110*n^3-16717449784653690*n^2+2358040920379200*n-146268798354000) *(n-2)^2/n^2/(2948689680*n^11-70667852760*n^10+761711962092*n^9-4874628375068*n ^8+20580890405304*n^7-60197562549050*n^6+124476682560457*n^5-181977361139651*n^ 4+184339059916975*n^3-123231781457679*n^2+48932930544400*n-8743942975500)/(3*n-\ 1)^2/(3*n-2)^2*b(n-3)-4*(2948689680*n^11-38232266280*n^10+217211366892*n^9-\ 712740293240*n^8+1498419303272*n^7-2115073845538*n^6+2040519733965*n^5-\ 1341682903844*n^4+587361445501*n^3-162521972848*n^2+25510883640*n-1720270800)*( 2*n-7)^2*(n-2)^2*(n-3)^2/n^2/(2948689680*n^11-70667852760*n^10+761711962092*n^9 -4874628375068*n^8+20580890405304*n^7-60197562549050*n^6+124476682560457*n^5-\ 181977361139651*n^4+184339059916975*n^3-123231781457679*n^2+48932930544400*n-\ 8743942975500)/(3*n-1)^2/(3*n-2)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 5, b(2) = 53, b(3) = 698 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 17 16 15 a(n) = 2/9 (23483364611520 n - 633248873215200 n + 7842268620985488 n 14 13 12 - 59176259116732588 n + 304375269420925548 n - 1131055969648436264 n 11 10 + 3139945515837906770 n - 6641753669827708749 n 9 8 + 10820005543539333064 n - 13630552859799932252 n 7 6 + 13256044178949879666 n - 9879674713109900653 n 5 4 3 + 5565183894263052808 n - 2316395259494917614 n + 687169983663160736 n 2 - 136767051939113880 n + 16293519136641600 n - 874779810912000) a(n - 1) / 2 2 2 17 / (n %1 (3 n - 1) (3 n - 2) ) - 4/9 (27817938441120 n / 16 15 14 - 777952276702320 n + 10030774054856688 n - 79141267090472332 n 13 12 + 427587074052228988 n - 1677251569760774892 n 11 10 + 4940837921091813374 n - 11150312161917911393 n 9 8 + 19487851717711047966 n - 26482263332780424296 n 7 6 + 27924525365091408845 n - 22665508662551929176 n 5 4 + 13949866511526698263 n - 6354066664650926851 n 3 2 + 2061398549040087156 n - 447186995298663540 n + 57710281016102400 n / 2 2 2 - 3327420122208000) a(n - 2) / (n %1 (3 n - 1) (3 n - 2) ) - 4/9 ( / 15 14 13 3031252991040 n - 75677805628320 n + 858344525581776 n 12 11 10 - 5857769424107608 n + 26854371245246884 n - 87440281467938624 n 9 8 7 + 208486612373084912 n - 369859713845792208 n + 491028173462289951 n 6 5 4 - 486556397579760371 n + 355858012913663087 n - 188041153939272139 n 3 2 + 69248122603406110 n - 16717449784653690 n + 2358040920379200 n 2 / 2 2 2 - 146268798354000) (n - 2) a(n - 3) / (n %1 (3 n - 1) (3 n - 2) ) - 4 / 11 10 9 8 (2948689680 n - 38232266280 n + 217211366892 n - 712740293240 n 7 6 5 + 1498419303272 n - 2115073845538 n + 2040519733965 n 4 3 2 - 1341682903844 n + 587361445501 n - 162521972848 n + 25510883640 n 2 2 2 / 2 2 - 1720270800) (2 n - 7) (n - 2) (n - 3) a(n - 4) / (n %1 (3 n - 1) / 2 (3 n - 2) ) 11 10 9 8 %1 := 2948689680 n - 70667852760 n + 761711962092 n - 4874628375068 n 7 6 5 + 20580890405304 n - 60197562549050 n + 124476682560457 n 4 3 2 - 181977361139651 n + 184339059916975 n - 123231781457679 n + 48932930544400 n - 8743942975500 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.59352100953272899805385962026306246672174646576686509838857397393978740551\ -1660 37451810209355403265836566 10 This constant is identified as, 0 The implied delta is, -0.6783578054091546747679095772754604999740396339229488\ 789130475779848087657110039663512460509477768583 Since this is negative, there is no Apery-style irrationality proof of, 0, but still a very fast way to compute it to many digits ----------------------- This took, 1.866, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(2 n + 2 k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(2 n + 2 k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 17 16 b(n) = 1/9 (18092532751141376 n - 486618153132971264 n 15 14 + 6010946479766827360 n - 45244349751504944000 n 13 12 + 232159998910923384088 n - 860769344658308017300 n 11 10 + 2384688910394248360004 n - 5035016691268019735592 n 9 8 + 8189837229356521345403 n - 10304594894886792723505 n 7 6 + 10012861734167487391394 n - 7459065466484449635158 n 5 4 + 4201427468669811049943 n - 1749371705889604329241 n 3 2 + 519336681564968787552 n - 103471023106252433460 n / 2 + 12342495716358796800 n - 663571233932544000) b(n - 1) / (n %1 / 2 2 17 (3 n - 1) (3 n - 2) ) - 1/9 (124922084678017792 n 16 15 - 3484836087193930880 n + 44815314175826729344 n 14 13 - 352616533341599963392 n + 1899650862197472616224 n 12 11 - 7429078372706530876920 n + 21815162558358065717552 n 10 9 - 49067647151442554734690 n + 85457101738370608550925 n 8 7 - 115701518017820633864504 n + 121533131882388912640983 n 6 5 - 98249644165570269276843 n + 60219590570355955405104 n 4 3 - 27314348514556139543031 n + 8824208833686845753756 n 2 - 1906446543992189648460 n + 245076970005096243840 n / 2 2 2 - 14079459260568844800) b(n - 2) / (n %1 (3 n - 1) (3 n - 2) ) - 1/9 ( / 15 14 13 7245276750557824 n - 180378967521101888 n + 2039419699026827552 n 12 11 - 13868843806419006608 n + 63330968461145913040 n 10 9 - 205321394276210699616 n + 487252205789598491980 n 8 7 - 860016324398529127738 n + 1135609198990487603347 n 6 5 - 1118909747089226696593 n + 813587138475014624093 n 4 3 - 427389877727891126799 n + 156482716684820503684 n 2 - 37568998970058448998 n + 5272190222361793920 n - 325511525123265600) 2 / 2 2 2 (n - 2) b(n - 3) / (n %1 (3 n - 1) (3 n - 2) ) - 49/9 ( / 11 10 9 607621330976 n - 7835931442000 n + 44258152291232 n 8 7 6 - 144319334255664 n + 301434372901332 n - 422662953495638 n 5 4 3 + 405067470679503 n - 264626965553536 n + 115143604431207 n 2 2 2 - 31682222097812 n + 4948430691680 n - 332219630400) (2 n - 7) (n - 2) 2 / 2 2 2 (n - 3) b(n - 4) / (n %1 (3 n - 1) (3 n - 2) ) / 11 10 9 %1 := 607621330976 n - 14519766082736 n + 156036639914912 n 8 7 6 - 995517139377792 n + 4190109341693076 n - 12217596373157954 n 5 4 3 + 25184952707888631 n - 36705052901566033 n + 37067925098331473 n 2 - 24705886598606845 n + 9781528428802872 n - 1742919278800980 and in Maple notation b(n) = 1/9*(18092532751141376*n^17-486618153132971264*n^16+6010946479766827360* n^15-45244349751504944000*n^14+232159998910923384088*n^13-860769344658308017300 *n^12+2384688910394248360004*n^11-5035016691268019735592*n^10+ 8189837229356521345403*n^9-10304594894886792723505*n^8+10012861734167487391394* n^7-7459065466484449635158*n^6+4201427468669811049943*n^5-\ 1749371705889604329241*n^4+519336681564968787552*n^3-103471023106252433460*n^2+ 12342495716358796800*n-663571233932544000)/n^2/(607621330976*n^11-\ 14519766082736*n^10+156036639914912*n^9-995517139377792*n^8+4190109341693076*n^ 7-12217596373157954*n^6+25184952707888631*n^5-36705052901566033*n^4+ 37067925098331473*n^3-24705886598606845*n^2+9781528428802872*n-1742919278800980 )/(3*n-1)^2/(3*n-2)^2*b(n-1)-1/9*(124922084678017792*n^17-3484836087193930880*n ^16+44815314175826729344*n^15-352616533341599963392*n^14+1899650862197472616224 *n^13-7429078372706530876920*n^12+21815162558358065717552*n^11-\ 49067647151442554734690*n^10+85457101738370608550925*n^9-\ 115701518017820633864504*n^8+121533131882388912640983*n^7-\ 98249644165570269276843*n^6+60219590570355955405104*n^5-27314348514556139543031 *n^4+8824208833686845753756*n^3-1906446543992189648460*n^2+ 245076970005096243840*n-14079459260568844800)/n^2/(607621330976*n^11-\ 14519766082736*n^10+156036639914912*n^9-995517139377792*n^8+4190109341693076*n^ 7-12217596373157954*n^6+25184952707888631*n^5-36705052901566033*n^4+ 37067925098331473*n^3-24705886598606845*n^2+9781528428802872*n-1742919278800980 )/(3*n-1)^2/(3*n-2)^2*b(n-2)-1/9*(7245276750557824*n^15-180378967521101888*n^14 +2039419699026827552*n^13-13868843806419006608*n^12+63330968461145913040*n^11-\ 205321394276210699616*n^10+487252205789598491980*n^9-860016324398529127738*n^8+ 1135609198990487603347*n^7-1118909747089226696593*n^6+813587138475014624093*n^5 -427389877727891126799*n^4+156482716684820503684*n^3-37568998970058448998*n^2+ 5272190222361793920*n-325511525123265600)*(n-2)^2/n^2/(607621330976*n^11-\ 14519766082736*n^10+156036639914912*n^9-995517139377792*n^8+4190109341693076*n^ 7-12217596373157954*n^6+25184952707888631*n^5-36705052901566033*n^4+ 37067925098331473*n^3-24705886598606845*n^2+9781528428802872*n-1742919278800980 )/(3*n-1)^2/(3*n-2)^2*b(n-3)-49/9*(607621330976*n^11-7835931442000*n^10+ 44258152291232*n^9-144319334255664*n^8+301434372901332*n^7-422662953495638*n^6+ 405067470679503*n^5-264626965553536*n^4+115143604431207*n^3-31682222097812*n^2+ 4948430691680*n-332219630400)*(2*n-7)^2*(n-2)^2*(n-3)^2/n^2/(607621330976*n^11-\ 14519766082736*n^10+156036639914912*n^9-995517139377792*n^8+4190109341693076*n^ 7-12217596373157954*n^6+25184952707888631*n^5-36705052901566033*n^4+ 37067925098331473*n^3-24705886598606845*n^2+9781528428802872*n-1742919278800980 )/(3*n-1)^2/(3*n-2)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 9, b(2) = 161, b(3) = 3525 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 17 16 a(n) = 1/9 (18092532751141376 n - 486618153132971264 n 15 14 + 6010946479766827360 n - 45244349751504944000 n 13 12 + 232159998910923384088 n - 860769344658308017300 n 11 10 + 2384688910394248360004 n - 5035016691268019735592 n 9 8 + 8189837229356521345403 n - 10304594894886792723505 n 7 6 + 10012861734167487391394 n - 7459065466484449635158 n 5 4 + 4201427468669811049943 n - 1749371705889604329241 n 3 2 + 519336681564968787552 n - 103471023106252433460 n / 2 + 12342495716358796800 n - 663571233932544000) a(n - 1) / (n %1 / 2 2 17 (3 n - 1) (3 n - 2) ) - 1/9 (124922084678017792 n 16 15 - 3484836087193930880 n + 44815314175826729344 n 14 13 - 352616533341599963392 n + 1899650862197472616224 n 12 11 - 7429078372706530876920 n + 21815162558358065717552 n 10 9 - 49067647151442554734690 n + 85457101738370608550925 n 8 7 - 115701518017820633864504 n + 121533131882388912640983 n 6 5 - 98249644165570269276843 n + 60219590570355955405104 n 4 3 - 27314348514556139543031 n + 8824208833686845753756 n 2 - 1906446543992189648460 n + 245076970005096243840 n / 2 2 2 - 14079459260568844800) a(n - 2) / (n %1 (3 n - 1) (3 n - 2) ) - 1/9 ( / 15 14 13 7245276750557824 n - 180378967521101888 n + 2039419699026827552 n 12 11 - 13868843806419006608 n + 63330968461145913040 n 10 9 - 205321394276210699616 n + 487252205789598491980 n 8 7 - 860016324398529127738 n + 1135609198990487603347 n 6 5 - 1118909747089226696593 n + 813587138475014624093 n 4 3 - 427389877727891126799 n + 156482716684820503684 n 2 - 37568998970058448998 n + 5272190222361793920 n - 325511525123265600) 2 / 2 2 2 (n - 2) a(n - 3) / (n %1 (3 n - 1) (3 n - 2) ) - 49/9 ( / 11 10 9 607621330976 n - 7835931442000 n + 44258152291232 n 8 7 6 - 144319334255664 n + 301434372901332 n - 422662953495638 n 5 4 3 + 405067470679503 n - 264626965553536 n + 115143604431207 n 2 2 2 - 31682222097812 n + 4948430691680 n - 332219630400) (2 n - 7) (n - 2) 2 / 2 2 2 (n - 3) a(n - 4) / (n %1 (3 n - 1) (3 n - 2) ) / 11 10 9 %1 := 607621330976 n - 14519766082736 n + 156036639914912 n 8 7 6 - 995517139377792 n + 4190109341693076 n - 12217596373157954 n 5 4 3 + 25184952707888631 n - 36705052901566033 n + 37067925098331473 n 2 - 24705886598606845 n + 9781528428802872 n - 1742919278800980 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.37808028030542471422758115715900540076845461874195741133855273245094197561\ 98334812395618147253192841 This constant is identified as, 6/11 ln(2) The implied delta is, -0.8013547722523136255126419149108569649659235985034022\ 581943453134271623041819783074145919492179874367 Since this is negative, there is no Apery-style irrationality proof of, 6/11 ln(2), but still a very fast way to compute it to many digits ----------------------- This took, 2.076, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(2 n + 2 k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(2 n + 2 k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 17 16 b(n) = 2/81 (2358935725899194304 n - 63394844959235863008 n 15 14 + 782460690934603085808 n - 5885024805565804833540 n 13 12 + 30175284220832370829276 n - 111802725921905799268700 n 11 10 + 309547068388448746161950 n - 653219804526804075763057 n 9 8 + 1062032263224920890139112 n - 1335808439758219420162320 n 7 6 + 1297701389096143298252942 n - 966631963658169724578001 n 5 4 + 544493879405538788832400 n - 226754586475544485248902 n 3 2 + 67337119988235867674496 n - 13421478190703416356600 n / 2 + 1601739167805021268800 n - 86158499011976160000) b(n - 1) / (n %1 / 2 2 17 (3 n - 1) (3 n - 2) ) - 4/81 (13646809316385760032 n 16 15 - 380395827232400485296 n + 4887949692980678853408 n 14 13 - 38426854998918670199964 n + 206834119838171887259724 n 12 11 - 808133299835978188541780 n + 2370765402540006181899322 n 10 9 - 5327074083933693413671873 n + 9267987333997677533512646 n 8 7 - 12534298449572909320036320 n + 13151028076735751708326749 n 6 5 - 10618964802103880966214604 n + 6500715249358826615292851 n 4 3 - 2944955964470072156536751 n + 950233523607024950713444 n 2 - 205050098157330957488964 n + 26329672929506336603136 n / 2 2 2 - 1511017193706858351360) b(n - 2) / (n %1 (3 n - 1) (3 n - 2) ) - / 15 14 4/81 (540486261895045824 n - 13444240011355346400 n 13 12 + 151855145579191827696 n - 1031538161531552781384 n 11 10 + 4704694718690048766780 n - 15232376040260007402608 n 9 8 + 36095625570676791867124 n - 63610288263418322878104 n 7 6 + 83854826334512819596323 n - 82478417235644316120063 n 5 4 + 59865349693729833075807 n - 31391938373792343651103 n 3 2 + 11473568065322055472622 n - 2750026809996291428082 n 2 / + 385327222956808126848 n - 23757340836022350480) (n - 2) b(n - 3) / ( / 2 2 2 484 11 n %1 (3 n - 1) (3 n - 2) ) - --- (12020422157616 n 81 10 9 8 - 154755023321208 n + 872471515410876 n - 2839448512077168 n 7 6 5 + 5918568585524432 n - 8281583070080114 n + 7920320642425985 n 4 3 2 - 5163792420885092 n + 2242539031453145 n - 615955265478560 n 2 2 2 + 96054730089624 n - 6439837821840) (2 n - 7) (n - 2) (n - 3) b(n - 4) / 2 2 2 / (n %1 (3 n - 1) (3 n - 2) ) / 11 10 9 %1 := 12020422157616 n - 286979667054984 n + 3081144967291836 n 8 7 6 - 19639037856236052 n + 82580473347491552 n - 240555718735697698 n 5 4 3 + 495391987890774533 n - 721297143174210943 n + 727733173459597355 n 2 - 484581851817698659 n + 191679440382489264 n - 34123949056725660 and in Maple notation b(n) = 2/81*(2358935725899194304*n^17-63394844959235863008*n^16+ 782460690934603085808*n^15-5885024805565804833540*n^14+30175284220832370829276* n^13-111802725921905799268700*n^12+309547068388448746161950*n^11-\ 653219804526804075763057*n^10+1062032263224920890139112*n^9-\ 1335808439758219420162320*n^8+1297701389096143298252942*n^7-\ 966631963658169724578001*n^6+544493879405538788832400*n^5-\ 226754586475544485248902*n^4+67337119988235867674496*n^3-\ 13421478190703416356600*n^2+1601739167805021268800*n-86158499011976160000)/n^2/ (12020422157616*n^11-286979667054984*n^10+3081144967291836*n^9-\ 19639037856236052*n^8+82580473347491552*n^7-240555718735697698*n^6+ 495391987890774533*n^5-721297143174210943*n^4+727733173459597355*n^3-\ 484581851817698659*n^2+191679440382489264*n-34123949056725660)/(3*n-1)^2/(3*n-2 )^2*b(n-1)-4/81*(13646809316385760032*n^17-380395827232400485296*n^16+ 4887949692980678853408*n^15-38426854998918670199964*n^14+ 206834119838171887259724*n^13-808133299835978188541780*n^12+ 2370765402540006181899322*n^11-5327074083933693413671873*n^10+ 9267987333997677533512646*n^9-12534298449572909320036320*n^8+ 13151028076735751708326749*n^7-10618964802103880966214604*n^6+ 6500715249358826615292851*n^5-2944955964470072156536751*n^4+ 950233523607024950713444*n^3-205050098157330957488964*n^2+ 26329672929506336603136*n-1511017193706858351360)/n^2/(12020422157616*n^11-\ 286979667054984*n^10+3081144967291836*n^9-19639037856236052*n^8+ 82580473347491552*n^7-240555718735697698*n^6+495391987890774533*n^5-\ 721297143174210943*n^4+727733173459597355*n^3-484581851817698659*n^2+ 191679440382489264*n-34123949056725660)/(3*n-1)^2/(3*n-2)^2*b(n-2)-4/81*( 540486261895045824*n^15-13444240011355346400*n^14+151855145579191827696*n^13-\ 1031538161531552781384*n^12+4704694718690048766780*n^11-15232376040260007402608 *n^10+36095625570676791867124*n^9-63610288263418322878104*n^8+ 83854826334512819596323*n^7-82478417235644316120063*n^6+59865349693729833075807 *n^5-31391938373792343651103*n^4+11473568065322055472622*n^3-\ 2750026809996291428082*n^2+385327222956808126848*n-23757340836022350480)*(n-2)^ 2/n^2/(12020422157616*n^11-286979667054984*n^10+3081144967291836*n^9-\ 19639037856236052*n^8+82580473347491552*n^7-240555718735697698*n^6+ 495391987890774533*n^5-721297143174210943*n^4+727733173459597355*n^3-\ 484581851817698659*n^2+191679440382489264*n-34123949056725660)/(3*n-1)^2/(3*n-2 )^2*b(n-3)-484/81*(12020422157616*n^11-154755023321208*n^10+872471515410876*n^9 -2839448512077168*n^8+5918568585524432*n^7-8281583070080114*n^6+ 7920320642425985*n^5-5163792420885092*n^4+2242539031453145*n^3-615955265478560* n^2+96054730089624*n-6439837821840)*(2*n-7)^2*(n-2)^2*(n-3)^2/n^2/( 12020422157616*n^11-286979667054984*n^10+3081144967291836*n^9-19639037856236052 *n^8+82580473347491552*n^7-240555718735697698*n^6+495391987890774533*n^5-\ 721297143174210943*n^4+727733173459597355*n^3-484581851817698659*n^2+ 191679440382489264*n-34123949056725660)/(3*n-1)^2/(3*n-2)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 13, b(2) = 325, b(3) = 9802 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 17 16 a(n) = 2/81 (2358935725899194304 n - 63394844959235863008 n 15 14 + 782460690934603085808 n - 5885024805565804833540 n 13 12 + 30175284220832370829276 n - 111802725921905799268700 n 11 10 + 309547068388448746161950 n - 653219804526804075763057 n 9 8 + 1062032263224920890139112 n - 1335808439758219420162320 n 7 6 + 1297701389096143298252942 n - 966631963658169724578001 n 5 4 + 544493879405538788832400 n - 226754586475544485248902 n 3 2 + 67337119988235867674496 n - 13421478190703416356600 n / 2 + 1601739167805021268800 n - 86158499011976160000) a(n - 1) / (n %1 / 2 2 17 (3 n - 1) (3 n - 2) ) - 4/81 (13646809316385760032 n 16 15 - 380395827232400485296 n + 4887949692980678853408 n 14 13 - 38426854998918670199964 n + 206834119838171887259724 n 12 11 - 808133299835978188541780 n + 2370765402540006181899322 n 10 9 - 5327074083933693413671873 n + 9267987333997677533512646 n 8 7 - 12534298449572909320036320 n + 13151028076735751708326749 n 6 5 - 10618964802103880966214604 n + 6500715249358826615292851 n 4 3 - 2944955964470072156536751 n + 950233523607024950713444 n 2 - 205050098157330957488964 n + 26329672929506336603136 n / 2 2 2 - 1511017193706858351360) a(n - 2) / (n %1 (3 n - 1) (3 n - 2) ) - / 15 14 4/81 (540486261895045824 n - 13444240011355346400 n 13 12 + 151855145579191827696 n - 1031538161531552781384 n 11 10 + 4704694718690048766780 n - 15232376040260007402608 n 9 8 + 36095625570676791867124 n - 63610288263418322878104 n 7 6 + 83854826334512819596323 n - 82478417235644316120063 n 5 4 + 59865349693729833075807 n - 31391938373792343651103 n 3 2 + 11473568065322055472622 n - 2750026809996291428082 n 2 / + 385327222956808126848 n - 23757340836022350480) (n - 2) a(n - 3) / ( / 2 2 2 484 11 n %1 (3 n - 1) (3 n - 2) ) - --- (12020422157616 n 81 10 9 8 - 154755023321208 n + 872471515410876 n - 2839448512077168 n 7 6 5 + 5918568585524432 n - 8281583070080114 n + 7920320642425985 n 4 3 2 - 5163792420885092 n + 2242539031453145 n - 615955265478560 n 2 2 2 + 96054730089624 n - 6439837821840) (2 n - 7) (n - 2) (n - 3) a(n - 4) / 2 2 2 / (n %1 (3 n - 1) (3 n - 2) ) / 11 10 9 %1 := 12020422157616 n - 286979667054984 n + 3081144967291836 n 8 7 6 - 19639037856236052 n + 82580473347491552 n - 240555718735697698 n 5 4 3 + 495391987890774533 n - 721297143174210943 n + 727733173459597355 n 2 - 484581851817698659 n + 191679440382489264 n - 34123949056725660 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.24413606414846882031005449709389460103277567951616654482993207414166539849\ 30242148608035010697182420 This constant is identified as, 2/9 ln(3) The implied delta is, -0.8480636789246993798140694984956357425546943949709526\ 251021911038449458544918368071979112958215773555 Since this is negative, there is no Apery-style irrationality proof of, 2/9 ln(3), but still a very fast way to compute it to many digits ----------------------- This took, 2.454, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ ) binomial(n, k) binomial(n + k, k) binomial(2 n + 2 k, k) / ----- k By Shalosh B. Ekhad Let ----- \ b(n) = ) binomial(n, k) binomial(n + k, k) binomial(2 n + 2 k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 12 11 10 9 b(n) = 2 (177681600 n - 3583245600 n + 31921648560 n - 165636440820 n 8 7 6 5 + 555762364704 n - 1265811007212 n + 1998716464360 n - 2194729747951 n 4 3 2 + 1655149213612 n - 831588135547 n + 262715987704 n - 46557144850 n / 2 12 + 3469502400) b(n - 1) / (n (3 n - 1) (3 n - 2) %1) + 4 (124377120 n / 11 10 9 8 - 2632649040 n + 24747705360 n - 136246588524 n + 487685705184 n 7 6 5 - 1190923925328 n + 2024460625736 n - 2399051656581 n 4 3 2 + 1952209618374 n - 1054388543289 n + 355274766311 n - 66346853633 n / 2 12 + 5059115040) b(n - 2) / (n (3 n - 1) (3 n - 2) %1) + 4 (11845440 n / 11 10 9 8 - 262573920 n + 2590213680 n - 14985980808 n + 56407914172 n 7 6 5 4 - 144814599572 n + 258463659532 n - 320780483878 n + 272339218409 n 3 2 - 152658728731 n + 53043224652 n - 10148295046 n + 787224480) b(n - 3) / 2 8 7 / (n (3 n - 1) (3 n - 2) %1) + 4/3 (987120 n - 10035720 n / 6 5 4 3 2 + 42172860 n - 94880864 n + 123757560 n - 94721666 n + 41112927 n 2 2 / 2 - 9222968 n + 793440) (n - 3) (2 n - 7) b(n - 4) / (n (3 n - 1) / (3 n - 2) %1) 8 7 6 5 4 %1 := 987120 n - 17932680 n + 140062260 n - 613946864 n + 1651103380 n 3 2 - 2788546666 n + 2887614305 n - 1676232540 n + 417685125 and in Maple notation b(n) = 2*(177681600*n^12-3583245600*n^11+31921648560*n^10-165636440820*n^9+ 555762364704*n^8-1265811007212*n^7+1998716464360*n^6-2194729747951*n^5+ 1655149213612*n^4-831588135547*n^3+262715987704*n^2-46557144850*n+3469502400)/n ^2/(3*n-1)/(3*n-2)/(987120*n^8-17932680*n^7+140062260*n^6-613946864*n^5+ 1651103380*n^4-2788546666*n^3+2887614305*n^2-1676232540*n+417685125)*b(n-1)+4*( 124377120*n^12-2632649040*n^11+24747705360*n^10-136246588524*n^9+487685705184*n ^8-1190923925328*n^7+2024460625736*n^6-2399051656581*n^5+1952209618374*n^4-\ 1054388543289*n^3+355274766311*n^2-66346853633*n+5059115040)/n^2/(3*n-1)/(3*n-2 )/(987120*n^8-17932680*n^7+140062260*n^6-613946864*n^5+1651103380*n^4-\ 2788546666*n^3+2887614305*n^2-1676232540*n+417685125)*b(n-2)+4*(11845440*n^12-\ 262573920*n^11+2590213680*n^10-14985980808*n^9+56407914172*n^8-144814599572*n^7 +258463659532*n^6-320780483878*n^5+272339218409*n^4-152658728731*n^3+ 53043224652*n^2-10148295046*n+787224480)/n^2/(3*n-1)/(3*n-2)/(987120*n^8-\ 17932680*n^7+140062260*n^6-613946864*n^5+1651103380*n^4-2788546666*n^3+ 2887614305*n^2-1676232540*n+417685125)*b(n-3)+4/3*(987120*n^8-10035720*n^7+ 42172860*n^6-94880864*n^5+123757560*n^4-94721666*n^3+41112927*n^2-9222968*n+ 793440)*(n-3)^2*(2*n-7)^2/n^2/(3*n-1)/(3*n-2)/(987120*n^8-17932680*n^7+ 140062260*n^6-613946864*n^5+1651103380*n^4-2788546666*n^3+2887614305*n^2-\ 1676232540*n+417685125)*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 9, b(2) = 205, b(3) = 5847 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 12 11 10 9 a(n) = 2 (177681600 n - 3583245600 n + 31921648560 n - 165636440820 n 8 7 6 5 + 555762364704 n - 1265811007212 n + 1998716464360 n - 2194729747951 n 4 3 2 + 1655149213612 n - 831588135547 n + 262715987704 n - 46557144850 n / 2 12 + 3469502400) a(n - 1) / (n (3 n - 1) (3 n - 2) %1) + 4 (124377120 n / 11 10 9 8 - 2632649040 n + 24747705360 n - 136246588524 n + 487685705184 n 7 6 5 - 1190923925328 n + 2024460625736 n - 2399051656581 n 4 3 2 + 1952209618374 n - 1054388543289 n + 355274766311 n - 66346853633 n / 2 12 + 5059115040) a(n - 2) / (n (3 n - 1) (3 n - 2) %1) + 4 (11845440 n / 11 10 9 8 - 262573920 n + 2590213680 n - 14985980808 n + 56407914172 n 7 6 5 4 - 144814599572 n + 258463659532 n - 320780483878 n + 272339218409 n 3 2 - 152658728731 n + 53043224652 n - 10148295046 n + 787224480) a(n - 3) / 2 8 7 / (n (3 n - 1) (3 n - 2) %1) + 4/3 (987120 n - 10035720 n / 6 5 4 3 2 + 42172860 n - 94880864 n + 123757560 n - 94721666 n + 41112927 n 2 2 / 2 - 9222968 n + 793440) (n - 3) (2 n - 7) a(n - 4) / (n (3 n - 1) / (3 n - 2) %1) 8 7 6 5 4 %1 := 987120 n - 17932680 n + 140062260 n - 613946864 n + 1651103380 n 3 2 - 2788546666 n + 2887614305 n - 1676232540 n + 417685125 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.10797426194481728723446423468729508139620973944887864620421798218676457282\ 61422947212229336540424749 The implied delta is, -0.4807817961072793312972636352796199567996961791969705\ 981004426002271490936885155378610576244663400939 Since this is negative, there is no Apery-style irrationality proof of, 0.107\ 974261944817287234464234687295081396209739448878646204217982186764572826\ 1422947212229336540424749, but still a very fast way to compute it to many digits ----------------------- This took, 5.405, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + k, k) binomial(2 n + 2 k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + k, k) binomial(2 n + 2 k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 12 11 10 b(n) = 1/3 (5256534528 n - 106006779648 n + 944450613920 n 9 8 7 - 4901494569120 n + 16450777877768 n - 37483540619604 n 6 5 4 + 59217782106360 n - 65069148660780 n + 49113984018709 n 3 2 - 24703523077305 n + 7815776000647 n - 1387781673315 n + 103678364160) / 2 12 b(n - 1) / (n (3 n - 1) (3 n - 2) %1) + 1/3 (2786358528 n / 11 10 9 - 58977922176 n + 554549695872 n - 3054760880512 n 8 7 6 + 10944788795872 n - 26765459466344 n + 45590445792680 n 5 4 3 - 54172067322618 n + 44236374310141 n - 23997721947708 n 2 / 2 + 8131580861618 n - 1531465881123 n + 119346474240) b(n - 2) / (n / 12 11 (3 n - 1) (3 n - 2) %1) + 1/3 (207494784 n - 4599467712 n 10 9 8 7 + 45362916960 n - 262352650352 n + 987016302272 n - 2532577777216 n 6 5 4 + 4517967268012 n - 5605707846258 n + 4759620563999 n 3 2 - 2669833962297 n + 929292477592 n - 178624549554 n + 14115588480) / 2 8 7 b(n - 3) / (n (3 n - 1) (3 n - 2) %1) + 1/3 (2470176 n - 25113456 n / 6 5 4 3 2 + 105568016 n - 237701120 n + 310504244 n - 238195006 n + 103708811 n 2 2 / 2 - 23387568 n + 2060160) (n - 3) (2 n - 7) b(n - 4) / (n (3 n - 1) / (3 n - 2) %1) 8 7 6 5 4 %1 := 2470176 n - 44874864 n + 350527136 n - 1536821648 n + 4134413364 n 3 2 - 6985884318 n + 7238398237 n - 4204876480 n + 1048708557 and in Maple notation b(n) = 1/3*(5256534528*n^12-106006779648*n^11+944450613920*n^10-4901494569120*n ^9+16450777877768*n^8-37483540619604*n^7+59217782106360*n^6-65069148660780*n^5+ 49113984018709*n^4-24703523077305*n^3+7815776000647*n^2-1387781673315*n+ 103678364160)/n^2/(3*n-1)/(3*n-2)/(2470176*n^8-44874864*n^7+350527136*n^6-\ 1536821648*n^5+4134413364*n^4-6985884318*n^3+7238398237*n^2-4204876480*n+ 1048708557)*b(n-1)+1/3*(2786358528*n^12-58977922176*n^11+554549695872*n^10-\ 3054760880512*n^9+10944788795872*n^8-26765459466344*n^7+45590445792680*n^6-\ 54172067322618*n^5+44236374310141*n^4-23997721947708*n^3+8131580861618*n^2-\ 1531465881123*n+119346474240)/n^2/(3*n-1)/(3*n-2)/(2470176*n^8-44874864*n^7+ 350527136*n^6-1536821648*n^5+4134413364*n^4-6985884318*n^3+7238398237*n^2-\ 4204876480*n+1048708557)*b(n-2)+1/3*(207494784*n^12-4599467712*n^11+45362916960 *n^10-262352650352*n^9+987016302272*n^8-2532577777216*n^7+4517967268012*n^6-\ 5605707846258*n^5+4759620563999*n^4-2669833962297*n^3+929292477592*n^2-\ 178624549554*n+14115588480)/n^2/(3*n-1)/(3*n-2)/(2470176*n^8-44874864*n^7+ 350527136*n^6-1536821648*n^5+4134413364*n^4-6985884318*n^3+7238398237*n^2-\ 4204876480*n+1048708557)*b(n-3)+1/3*(2470176*n^8-25113456*n^7+105568016*n^6-\ 237701120*n^5+310504244*n^4-238195006*n^3+103708811*n^2-23387568*n+2060160)*(n-\ 3)^2*(2*n-7)^2/n^2/(3*n-1)/(3*n-2)/(2470176*n^8-44874864*n^7+350527136*n^6-\ 1536821648*n^5+4134413364*n^4-6985884318*n^3+7238398237*n^2-4204876480*n+ 1048708557)*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 17, b(2) = 745, b(3) = 40793 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 12 11 10 a(n) = 1/3 (5256534528 n - 106006779648 n + 944450613920 n 9 8 7 - 4901494569120 n + 16450777877768 n - 37483540619604 n 6 5 4 + 59217782106360 n - 65069148660780 n + 49113984018709 n 3 2 - 24703523077305 n + 7815776000647 n - 1387781673315 n + 103678364160) / 2 12 a(n - 1) / (n (3 n - 1) (3 n - 2) %1) + 1/3 (2786358528 n / 11 10 9 - 58977922176 n + 554549695872 n - 3054760880512 n 8 7 6 + 10944788795872 n - 26765459466344 n + 45590445792680 n 5 4 3 - 54172067322618 n + 44236374310141 n - 23997721947708 n 2 / 2 + 8131580861618 n - 1531465881123 n + 119346474240) a(n - 2) / (n / 12 11 (3 n - 1) (3 n - 2) %1) + 1/3 (207494784 n - 4599467712 n 10 9 8 7 + 45362916960 n - 262352650352 n + 987016302272 n - 2532577777216 n 6 5 4 + 4517967268012 n - 5605707846258 n + 4759620563999 n 3 2 - 2669833962297 n + 929292477592 n - 178624549554 n + 14115588480) / 2 8 7 a(n - 3) / (n (3 n - 1) (3 n - 2) %1) + 1/3 (2470176 n - 25113456 n / 6 5 4 3 2 + 105568016 n - 237701120 n + 310504244 n - 238195006 n + 103708811 n 2 2 / 2 - 23387568 n + 2060160) (n - 3) (2 n - 7) a(n - 4) / (n (3 n - 1) / (3 n - 2) %1) 8 7 6 5 4 %1 := 2470176 n - 44874864 n + 350527136 n - 1536821648 n + 4134413364 n 3 2 - 6985884318 n + 7238398237 n - 4204876480 n + 1048708557 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.05846355282424832904966496043275291207821066409728334671985768900850474877\ 348392502316928006762184608 7232 This constant is identified as, ----- ln(2) 85743 The implied delta is, -0.4085139225203998850639468721750085492974607039826882\ 081229693556362255488006979246998318769922634482 Since this is negative, there is no Apery-style irrationality proof of, 7232 ----- ln(2), but still a very fast way to compute it to many digits 85743 ----------------------- This took, 5.426, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + k, k) binomial(2 n + 2 k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + k, k) binomial(2 n + 2 k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 12 11 10 b(n) = 2/27 (349508098368 n - 7048413317088 n + 62798668380144 n 9 8 7 - 325933011591988 n + 1094035605896056 n - 2493150724091288 n 6 5 4 + 3939510428240836 n - 4329820091652365 n + 3269137797047608 n 3 2 - 1644978086820253 n + 520720850045568 n - 92528674644078 n / 2 + 6920097468480) b(n - 1) / (n (3 n - 1) (3 n - 2) %1) + 4/9 ( / 12 11 10 9 18542717856 n - 392487527952 n + 3690820529760 n - 20335892366284 n 8 7 6 + 72889922949208 n - 178358536876984 n + 304055121811564 n 5 4 3 - 361682271242109 n + 295750920826450 n - 160703018316209 n 2 / 2 + 54553118025003 n - 10296221473613 n + 806253950880) b(n - 2) / (n / 12 11 (3 n - 1) (3 n - 2) %1) + 4/9 (1082068416 n - 23985849888 n 10 9 8 + 236551442448 n - 1367947091096 n + 5145844261092 n 7 6 5 - 13202081372924 n + 23549461379736 n - 29218075280302 n 4 3 2 + 24809315235249 n - 13918519524463 n + 4845932283332 n / 2 - 932076918430 n + 73932717600) b(n - 3) / (n (3 n - 1) (3 n - 2) %1) / 8 7 6 5 + 4/27 (24592464 n - 250023384 n + 1051124796 n - 2367390424 n 4 3 2 + 3093926160 n - 2374979794 n + 1034777043 n - 233524264 n + 20671200) 2 2 / 2 (n - 3) (2 n - 7) b(n - 4) / (n (3 n - 1) (3 n - 2) %1) / 8 7 6 5 %1 := 24592464 n - 446763096 n + 3489877476 n - 15301808248 n 4 3 2 + 41170041140 n - 69575081018 n + 72103129621 n - 41894326668 n + 10451009529 and in Maple notation b(n) = 2/27*(349508098368*n^12-7048413317088*n^11+62798668380144*n^10-\ 325933011591988*n^9+1094035605896056*n^8-2493150724091288*n^7+3939510428240836* n^6-4329820091652365*n^5+3269137797047608*n^4-1644978086820253*n^3+ 520720850045568*n^2-92528674644078*n+6920097468480)/n^2/(3*n-1)/(3*n-2)/( 24592464*n^8-446763096*n^7+3489877476*n^6-15301808248*n^5+41170041140*n^4-\ 69575081018*n^3+72103129621*n^2-41894326668*n+10451009529)*b(n-1)+4/9*( 18542717856*n^12-392487527952*n^11+3690820529760*n^10-20335892366284*n^9+ 72889922949208*n^8-178358536876984*n^7+304055121811564*n^6-361682271242109*n^5+ 295750920826450*n^4-160703018316209*n^3+54553118025003*n^2-10296221473613*n+ 806253950880)/n^2/(3*n-1)/(3*n-2)/(24592464*n^8-446763096*n^7+3489877476*n^6-\ 15301808248*n^5+41170041140*n^4-69575081018*n^3+72103129621*n^2-41894326668*n+ 10451009529)*b(n-2)+4/9*(1082068416*n^12-23985849888*n^11+236551442448*n^10-\ 1367947091096*n^9+5145844261092*n^8-13202081372924*n^7+23549461379736*n^6-\ 29218075280302*n^5+24809315235249*n^4-13918519524463*n^3+4845932283332*n^2-\ 932076918430*n+73932717600)/n^2/(3*n-1)/(3*n-2)/(24592464*n^8-446763096*n^7+ 3489877476*n^6-15301808248*n^5+41170041140*n^4-69575081018*n^3+72103129621*n^2-\ 41894326668*n+10451009529)*b(n-3)+4/27*(24592464*n^8-250023384*n^7+1051124796*n ^6-2367390424*n^5+3093926160*n^4-2374979794*n^3+1034777043*n^2-233524264*n+ 20671200)*(n-3)^2*(2*n-7)^2/n^2/(3*n-1)/(3*n-2)/(24592464*n^8-446763096*n^7+ 3489877476*n^6-15301808248*n^5+41170041140*n^4-69575081018*n^3+72103129621*n^2-\ 41894326668*n+10451009529)*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 25, b(2) = 1621, b(3) = 131239 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 12 11 10 a(n) = 2/27 (349508098368 n - 7048413317088 n + 62798668380144 n 9 8 7 - 325933011591988 n + 1094035605896056 n - 2493150724091288 n 6 5 4 + 3939510428240836 n - 4329820091652365 n + 3269137797047608 n 3 2 - 1644978086820253 n + 520720850045568 n - 92528674644078 n / 2 + 6920097468480) a(n - 1) / (n (3 n - 1) (3 n - 2) %1) + 4/9 ( / 12 11 10 9 18542717856 n - 392487527952 n + 3690820529760 n - 20335892366284 n 8 7 6 + 72889922949208 n - 178358536876984 n + 304055121811564 n 5 4 3 - 361682271242109 n + 295750920826450 n - 160703018316209 n 2 / 2 + 54553118025003 n - 10296221473613 n + 806253950880) a(n - 2) / (n / 12 11 (3 n - 1) (3 n - 2) %1) + 4/9 (1082068416 n - 23985849888 n 10 9 8 + 236551442448 n - 1367947091096 n + 5145844261092 n 7 6 5 - 13202081372924 n + 23549461379736 n - 29218075280302 n 4 3 2 + 24809315235249 n - 13918519524463 n + 4845932283332 n / 2 - 932076918430 n + 73932717600) a(n - 3) / (n (3 n - 1) (3 n - 2) %1) / 8 7 6 5 + 4/27 (24592464 n - 250023384 n + 1051124796 n - 2367390424 n 4 3 2 + 3093926160 n - 2374979794 n + 1034777043 n - 233524264 n + 20671200) 2 2 / 2 (n - 3) (2 n - 7) a(n - 4) / (n (3 n - 1) (3 n - 2) %1) / 8 7 6 5 %1 := 24592464 n - 446763096 n + 3489877476 n - 15301808248 n 4 3 2 + 41170041140 n - 69575081018 n + 72103129621 n - 41894326668 n + 10451009529 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.03989846749407357018501950276168786773599726272300818869750196531616513157\ 617454101909415482527895688 The implied delta is, -0.3829191213060336346358215568378287000110470872875984\ 733572223157281457396784769951472749240429191337 Since this is negative, there is no Apery-style irrationality proof of, 0.039\ 898467494073570185019502761687867735997262723008188697501965316165131576\ 17454101909415482527895688, but still a very fast way to compute it to many digits ----------------------- This took, 6.946, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n + 2 k, k) / ----- k By Shalosh B. Ekhad Let ----- \ b(n) = ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n + 2 k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 16 15 14 b(n) = 2/9 (33762201421312 n - 816920307486976 n + 8997390452139152 n 13 12 11 - 59796865467273916 n + 268037933923322652 n - 857984624899860028 n 10 9 + 2025523340108361302 n - 3591252667466616947 n 8 7 6 + 4823625073753396606 n - 4914401046255353108 n + 3776861223446589510 n 5 4 3 - 2161473025663642715 n + 900863361811070886 n - 263724933147858102 n 2 + 51038748826056612 n - 5826058176273120 n + 295365716956800) b(n - 1) / 2 2 2 15 / (n %1 (3 n - 1) (3 n - 2) ) - 4/9 (n - 1) (5804952554336 n / 14 13 12 - 140458365450128 n + 1541135769882768 n - 10152843084969084 n 11 10 9 + 44815804713023620 n - 140053878874610312 n + 319196237263316094 n 8 7 6 - 538467183774724759 n + 675374547608327616 n - 627246299635751736 n 5 4 3 + 426000872535796229 n - 206899294881547855 n + 69302107706713429 n 2 - 15071642696443602 n + 1899929455434144 n - 104599802926080) b(n - 2) / 2 2 2 14 / (n %1 (3 n - 1) (3 n - 2) ) + 4/9 (n - 1) (n - 2) (647145734976 n / 13 12 11 - 15011385715872 n + 156391937548848 n - 967290375920376 n 10 9 8 + 3955739396601012 n - 11275235914709756 n + 23008986795620672 n 7 6 5 - 33999849652639084 n + 36384702082358675 n - 27923993590659244 n 4 3 2 + 15058261062908363 n - 5511002105222216 n + 1290536221144506 n / 2 2 - 172945025878464 n + 10015069463280) b(n - 3) / (n %1 (3 n - 1) / 2 10 9 (3 n - 2) ) - 68/9 (n - 1) (n - 2) (1382790032 n - 15482130120 n 8 7 6 5 + 74002100540 n - 197832708140 n + 325904041804 n - 344111505362 n 4 3 2 + 235099547825 n - 102539756111 n + 27356653436 n - 4040740584 n 2 2 / 2 2 2 + 251516880) (n - 3) (2 n - 7) b(n - 4) / (n %1 (3 n - 1) (3 n - 2) / ) 10 9 8 7 %1 := 1382790032 n - 29310030440 n + 275566823060 n - 1513141000620 n 6 5 4 + 5373676650704 n - 12897351740550 n + 21189643826235 n 3 2 - 23536829986171 n + 16921376811219 n - 7112766117423 n + 1328003490834 and in Maple notation b(n) = 2/9*(33762201421312*n^16-816920307486976*n^15+8997390452139152*n^14-\ 59796865467273916*n^13+268037933923322652*n^12-857984624899860028*n^11+ 2025523340108361302*n^10-3591252667466616947*n^9+4823625073753396606*n^8-\ 4914401046255353108*n^7+3776861223446589510*n^6-2161473025663642715*n^5+ 900863361811070886*n^4-263724933147858102*n^3+51038748826056612*n^2-\ 5826058176273120*n+295365716956800)/n^2/(1382790032*n^10-29310030440*n^9+ 275566823060*n^8-1513141000620*n^7+5373676650704*n^6-12897351740550*n^5+ 21189643826235*n^4-23536829986171*n^3+16921376811219*n^2-7112766117423*n+ 1328003490834)/(3*n-1)^2/(3*n-2)^2*b(n-1)-4/9*(n-1)*(5804952554336*n^15-\ 140458365450128*n^14+1541135769882768*n^13-10152843084969084*n^12+ 44815804713023620*n^11-140053878874610312*n^10+319196237263316094*n^9-\ 538467183774724759*n^8+675374547608327616*n^7-627246299635751736*n^6+ 426000872535796229*n^5-206899294881547855*n^4+69302107706713429*n^3-\ 15071642696443602*n^2+1899929455434144*n-104599802926080)/n^2/(1382790032*n^10-\ 29310030440*n^9+275566823060*n^8-1513141000620*n^7+5373676650704*n^6-\ 12897351740550*n^5+21189643826235*n^4-23536829986171*n^3+16921376811219*n^2-\ 7112766117423*n+1328003490834)/(3*n-1)^2/(3*n-2)^2*b(n-2)+4/9*(n-1)*(n-2)*( 647145734976*n^14-15011385715872*n^13+156391937548848*n^12-967290375920376*n^11 +3955739396601012*n^10-11275235914709756*n^9+23008986795620672*n^8-\ 33999849652639084*n^7+36384702082358675*n^6-27923993590659244*n^5+ 15058261062908363*n^4-5511002105222216*n^3+1290536221144506*n^2-172945025878464 *n+10015069463280)/n^2/(1382790032*n^10-29310030440*n^9+275566823060*n^8-\ 1513141000620*n^7+5373676650704*n^6-12897351740550*n^5+21189643826235*n^4-\ 23536829986171*n^3+16921376811219*n^2-7112766117423*n+1328003490834)/(3*n-1)^2/ (3*n-2)^2*b(n-3)-68/9*(n-1)*(n-2)*(1382790032*n^10-15482130120*n^9+74002100540* n^8-197832708140*n^7+325904041804*n^6-344111505362*n^5+235099547825*n^4-\ 102539756111*n^3+27356653436*n^2-4040740584*n+251516880)*(n-3)^2*(2*n-7)^2/n^2/ (1382790032*n^10-29310030440*n^9+275566823060*n^8-1513141000620*n^7+ 5373676650704*n^6-12897351740550*n^5+21189643826235*n^4-23536829986171*n^3+ 16921376811219*n^2-7112766117423*n+1328003490834)/(3*n-1)^2/(3*n-2)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 13, b(2) = 469, b(3) = 21436 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 16 15 14 a(n) = 2/9 (33762201421312 n - 816920307486976 n + 8997390452139152 n 13 12 11 - 59796865467273916 n + 268037933923322652 n - 857984624899860028 n 10 9 + 2025523340108361302 n - 3591252667466616947 n 8 7 6 + 4823625073753396606 n - 4914401046255353108 n + 3776861223446589510 n 5 4 3 - 2161473025663642715 n + 900863361811070886 n - 263724933147858102 n 2 + 51038748826056612 n - 5826058176273120 n + 295365716956800) a(n - 1) / 2 2 2 15 / (n %1 (3 n - 1) (3 n - 2) ) - 4/9 (n - 1) (5804952554336 n / 14 13 12 - 140458365450128 n + 1541135769882768 n - 10152843084969084 n 11 10 9 + 44815804713023620 n - 140053878874610312 n + 319196237263316094 n 8 7 6 - 538467183774724759 n + 675374547608327616 n - 627246299635751736 n 5 4 3 + 426000872535796229 n - 206899294881547855 n + 69302107706713429 n 2 - 15071642696443602 n + 1899929455434144 n - 104599802926080) a(n - 2) / 2 2 2 14 / (n %1 (3 n - 1) (3 n - 2) ) + 4/9 (n - 1) (n - 2) (647145734976 n / 13 12 11 - 15011385715872 n + 156391937548848 n - 967290375920376 n 10 9 8 + 3955739396601012 n - 11275235914709756 n + 23008986795620672 n 7 6 5 - 33999849652639084 n + 36384702082358675 n - 27923993590659244 n 4 3 2 + 15058261062908363 n - 5511002105222216 n + 1290536221144506 n / 2 2 - 172945025878464 n + 10015069463280) a(n - 3) / (n %1 (3 n - 1) / 2 10 9 (3 n - 2) ) - 68/9 (n - 1) (n - 2) (1382790032 n - 15482130120 n 8 7 6 5 + 74002100540 n - 197832708140 n + 325904041804 n - 344111505362 n 4 3 2 + 235099547825 n - 102539756111 n + 27356653436 n - 4040740584 n 2 2 / 2 2 2 + 251516880) (n - 3) (2 n - 7) a(n - 4) / (n %1 (3 n - 1) (3 n - 2) / ) 10 9 8 7 %1 := 1382790032 n - 29310030440 n + 275566823060 n - 1513141000620 n 6 5 4 + 5373676650704 n - 12897351740550 n + 21189643826235 n 3 2 - 23536829986171 n + 16921376811219 n - 7112766117423 n + 1328003490834 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.07708381652240925625474442639460593147242390560621096547108088786325613345\ 282431212802677138601275334 The implied delta is, -0.4618025872883926827040984707277121654795099380869658\ 832865755541673641771037504863577383976009771478 Since this is negative, there is no Apery-style irrationality proof of, 0.077\ 083816522409256254744426394605931472423905606210965471080887863256133452\ 82431212802677138601275334, but still a very fast way to compute it to many digits ----------------------- This took, 7.165, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n + 2 k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n + 2 k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 16 15 b(n) = 1/9 (2931875264071936 n - 70996727248859776 n 14 13 + 782900977718382368 n - 5211515988363274464 n 12 11 + 23405375851066844184 n - 75083611632941438372 n 10 9 + 177679305835463788412 n - 315823215249078760256 n 8 7 + 425318523264177504823 n - 434491419483972382887 n 6 5 + 334830686919897012492 n - 192145780457661350278 n 4 3 + 80300843103331008567 n - 23570686017671866815 n 2 + 4573356190687976706 n - 523272726475626240 n + 26579699543577600) / 2 2 2 b(n - 1) / (n %1 (3 n - 1) (3 n - 2) ) - 1/9 (n - 1) ( / 15 14 13 490162520031488 n - 11869514084974208 n + 130403114453252608 n 12 11 - 860672203341422080 n + 3808472004513144768 n 10 9 - 11939233843082047144 n + 27316043028174788952 n 8 7 - 46295069991652855354 n + 58382239736212753991 n 6 5 - 54558880050359773391 n + 37308353414302913736 n 4 3 - 18251034604924906721 n + 6157371972444209401 n 2 - 1348095120913792686 n + 170965070434992960 n - 9465504716908800) / 2 2 2 b(n - 2) / (n %1 (3 n - 1) (3 n - 2) ) + 1/9 (n - 1) (n - 2) ( / 14 13 12 44146599270272 n - 1024883896591680 n + 10694771982733472 n 11 10 9 - 66310230705881488 n + 272082809620288800 n - 778844130747997408 n 8 7 6 + 1597669829524362604 n - 2375416596393632034 n + 2559939427023353143 n 5 4 3 - 1979839623091932020 n + 1076253224937873915 n - 396995372940654626 n 2 + 93633986593513770 n - 12625259978931840 n + 734962256683200) b(n - 3) / 2 2 2 10 / (n %1 (3 n - 1) (3 n - 2) ) - 11/3 (n - 1) (n - 2) (30742757152 n / 9 8 7 6 - 344794363056 n + 1654534091920 n - 4450335700224 n + 7391714354316 n 5 4 3 2 - 7882409508282 n + 5444433020297 n - 2400415331703 n + 646576538972 n 2 2 / 2 - 96277425696 n + 6037485120) (n - 3) (2 n - 7) b(n - 4) / (n %1 / 2 2 (3 n - 1) (3 n - 2) ) 10 9 8 7 %1 := 30742757152 n - 652221934576 n + 6141107431264 n - 33788336363840 n 6 5 4 + 120289724328268 n - 289534919033762 n + 477211400565663 n 3 2 - 531904045592335 n + 383794256473743 n - 161929941723195 n + 30348270576738 and in Maple notation b(n) = 1/9*(2931875264071936*n^16-70996727248859776*n^15+782900977718382368*n^ 14-5211515988363274464*n^13+23405375851066844184*n^12-75083611632941438372*n^11 +177679305835463788412*n^10-315823215249078760256*n^9+425318523264177504823*n^8 -434491419483972382887*n^7+334830686919897012492*n^6-192145780457661350278*n^5+ 80300843103331008567*n^4-23570686017671866815*n^3+4573356190687976706*n^2-\ 523272726475626240*n+26579699543577600)/n^2/(30742757152*n^10-652221934576*n^9+ 6141107431264*n^8-33788336363840*n^7+120289724328268*n^6-289534919033762*n^5+ 477211400565663*n^4-531904045592335*n^3+383794256473743*n^2-161929941723195*n+ 30348270576738)/(3*n-1)^2/(3*n-2)^2*b(n-1)-1/9*(n-1)*(490162520031488*n^15-\ 11869514084974208*n^14+130403114453252608*n^13-860672203341422080*n^12+ 3808472004513144768*n^11-11939233843082047144*n^10+27316043028174788952*n^9-\ 46295069991652855354*n^8+58382239736212753991*n^7-54558880050359773391*n^6+ 37308353414302913736*n^5-18251034604924906721*n^4+6157371972444209401*n^3-\ 1348095120913792686*n^2+170965070434992960*n-9465504716908800)/n^2/(30742757152 *n^10-652221934576*n^9+6141107431264*n^8-33788336363840*n^7+120289724328268*n^6 -289534919033762*n^5+477211400565663*n^4-531904045592335*n^3+383794256473743*n^ 2-161929941723195*n+30348270576738)/(3*n-1)^2/(3*n-2)^2*b(n-2)+1/9*(n-1)*(n-2)* (44146599270272*n^14-1024883896591680*n^13+10694771982733472*n^12-\ 66310230705881488*n^11+272082809620288800*n^10-778844130747997408*n^9+ 1597669829524362604*n^8-2375416596393632034*n^7+2559939427023353143*n^6-\ 1979839623091932020*n^5+1076253224937873915*n^4-396995372940654626*n^3+ 93633986593513770*n^2-12625259978931840*n+734962256683200)/n^2/(30742757152*n^ 10-652221934576*n^9+6141107431264*n^8-33788336363840*n^7+120289724328268*n^6-\ 289534919033762*n^5+477211400565663*n^4-531904045592335*n^3+383794256473743*n^2 -161929941723195*n+30348270576738)/(3*n-1)^2/(3*n-2)^2*b(n-3)-11/3*(n-1)*(n-2)* (30742757152*n^10-344794363056*n^9+1654534091920*n^8-4450335700224*n^7+ 7391714354316*n^6-7882409508282*n^5+5444433020297*n^4-2400415331703*n^3+ 646576538972*n^2-96277425696*n+6037485120)*(n-3)^2*(2*n-7)^2/n^2/(30742757152*n ^10-652221934576*n^9+6141107431264*n^8-33788336363840*n^7+120289724328268*n^6-\ 289534919033762*n^5+477211400565663*n^4-531904045592335*n^3+383794256473743*n^2 -161929941723195*n+30348270576738)/(3*n-1)^2/(3*n-2)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 25, b(2) = 1777, b(3) = 159421 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 16 15 a(n) = 1/9 (2931875264071936 n - 70996727248859776 n 14 13 + 782900977718382368 n - 5211515988363274464 n 12 11 + 23405375851066844184 n - 75083611632941438372 n 10 9 + 177679305835463788412 n - 315823215249078760256 n 8 7 + 425318523264177504823 n - 434491419483972382887 n 6 5 + 334830686919897012492 n - 192145780457661350278 n 4 3 + 80300843103331008567 n - 23570686017671866815 n 2 + 4573356190687976706 n - 523272726475626240 n + 26579699543577600) / 2 2 2 a(n - 1) / (n %1 (3 n - 1) (3 n - 2) ) - 1/9 (n - 1) ( / 15 14 13 490162520031488 n - 11869514084974208 n + 130403114453252608 n 12 11 - 860672203341422080 n + 3808472004513144768 n 10 9 - 11939233843082047144 n + 27316043028174788952 n 8 7 - 46295069991652855354 n + 58382239736212753991 n 6 5 - 54558880050359773391 n + 37308353414302913736 n 4 3 - 18251034604924906721 n + 6157371972444209401 n 2 - 1348095120913792686 n + 170965070434992960 n - 9465504716908800) / 2 2 2 a(n - 2) / (n %1 (3 n - 1) (3 n - 2) ) + 1/9 (n - 1) (n - 2) ( / 14 13 12 44146599270272 n - 1024883896591680 n + 10694771982733472 n 11 10 9 - 66310230705881488 n + 272082809620288800 n - 778844130747997408 n 8 7 6 + 1597669829524362604 n - 2375416596393632034 n + 2559939427023353143 n 5 4 3 - 1979839623091932020 n + 1076253224937873915 n - 396995372940654626 n 2 + 93633986593513770 n - 12625259978931840 n + 734962256683200) a(n - 3) / 2 2 2 10 / (n %1 (3 n - 1) (3 n - 2) ) - 11/3 (n - 1) (n - 2) (30742757152 n / 9 8 7 6 - 344794363056 n + 1654534091920 n - 4450335700224 n + 7391714354316 n 5 4 3 2 - 7882409508282 n + 5444433020297 n - 2400415331703 n + 646576538972 n 2 2 / 2 - 96277425696 n + 6037485120) (n - 3) (2 n - 7) a(n - 4) / (n %1 / 2 2 (3 n - 1) (3 n - 2) ) 10 9 8 7 %1 := 30742757152 n - 652221934576 n + 6141107431264 n - 33788336363840 n 6 5 4 + 120289724328268 n - 289534919033762 n + 477211400565663 n 3 2 - 531904045592335 n + 383794256473743 n - 161929941723195 n + 30348270576738 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.04002885373269341284835049125539513227961373321197292129471648116329789107\ 982057295506735987358884521 The implied delta is, -0.4356198621891237697104486967816099450959577469259498\ 238576048017883463841163905712534418629269865396 Since this is negative, there is no Apery-style irrationality proof of, 0.040\ 028853732693412848350491255395132279613733211972921294716481163297891079\ 82057295506735987358884521, but still a very fast way to compute it to many digits ----------------------- This took, 5.503, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n + 2 k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(n + 2 k, k) binomial(2 n + 2 k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 16 15 b(n) = 2/81 (128497239609398400 n - 3112761205396939200 n 14 13 + 34345417290576238800 n - 228803562403738788260 n 12 11 + 1028535770673700906092 n - 3302997028521842942488 n 10 9 + 7825272925840021644042 n - 13926235910289347242131 n 8 7 + 18777894159821846304942 n - 19207136041572044888704 n 6 5 + 14820174797314619699274 n - 8515242416561157904907 n 4 3 + 3562955073218115434190 n - 1047047813800287639030 n 2 + 203378003954398288260 n - 23292999112440123360 n + 1184133976143350400) / 2 2 2 b(n - 1) / (n %1 (3 n - 1) (3 n - 2) ) - 4/81 (n - 1) ( / 15 14 13 7089172775100000 n - 171730552810050000 n + 1887864001985318400 n 12 11 - 12471216714280649820 n + 55251258197116303740 n 10 9 - 173473154113028450800 n + 397642988067484830182 n 8 7 - 675451472451117045855 n + 854066396315000315160 n 6 5 - 800550325162481208888 n + 549260201123119756341 n 4 3 - 269646346685451368015 n + 91294144805520534301 n 2 - 20054748000835577442 n + 2550882268357014816 n - 141591568451397120) / 2 2 2 b(n - 2) / (n %1 (3 n - 1) (3 n - 2) ) + 4/81 (n - 1) (n - 2) ( / 14 13 12 586440505310400 n - 13619695437784800 n + 142231711903750800 n 11 10 - 882882552472392360 n + 3628148568465988572 n 9 8 - 10405371442147524020 n + 21393063306892695308 n 7 6 - 31889587650802556632 n + 34465531413211968071 n 5 4 - 26737365735758218336 n + 14580360549067214051 n 3 2 - 5394625116788678624 n + 1275871055962718202 n - 172435742329318272 n / 2 2 2 196 + 10056781528118640) b(n - 3) / (n %1 (3 n - 1) (3 n - 2) ) - --- / 81 10 9 8 (n - 1) (n - 2) (201111284400 n - 2257342035000 n + 10852926995100 n 7 6 5 - 29280832962820 n + 48831757675652 n - 52329773348458 n 4 3 2 + 36340723184225 n - 16109138683447 n + 4360283292188 n 2 2 / 2 - 651903197544 n + 41018147280) (n - 3) (2 n - 7) b(n - 4) / (n %1 / 2 2 (3 n - 1) (3 n - 2) ) 10 9 8 %1 := 201111284400 n - 4268454879000 n + 40219013108100 n 7 6 5 - 221501916311620 n + 789529644942192 n - 1903086863425990 n 4 3 2 + 3141658464550995 n - 3507748068910267 n + 2535599906207259 n - 1071818629224903 n + 201256810806114 and in Maple notation b(n) = 2/81*(128497239609398400*n^16-3112761205396939200*n^15+ 34345417290576238800*n^14-228803562403738788260*n^13+1028535770673700906092*n^ 12-3302997028521842942488*n^11+7825272925840021644042*n^10-\ 13926235910289347242131*n^9+18777894159821846304942*n^8-19207136041572044888704 *n^7+14820174797314619699274*n^6-8515242416561157904907*n^5+ 3562955073218115434190*n^4-1047047813800287639030*n^3+203378003954398288260*n^2 -23292999112440123360*n+1184133976143350400)/n^2/(201111284400*n^10-\ 4268454879000*n^9+40219013108100*n^8-221501916311620*n^7+789529644942192*n^6-\ 1903086863425990*n^5+3141658464550995*n^4-3507748068910267*n^3+2535599906207259 *n^2-1071818629224903*n+201256810806114)/(3*n-1)^2/(3*n-2)^2*b(n-1)-4/81*(n-1)* (7089172775100000*n^15-171730552810050000*n^14+1887864001985318400*n^13-\ 12471216714280649820*n^12+55251258197116303740*n^11-173473154113028450800*n^10+ 397642988067484830182*n^9-675451472451117045855*n^8+854066396315000315160*n^7-\ 800550325162481208888*n^6+549260201123119756341*n^5-269646346685451368015*n^4+ 91294144805520534301*n^3-20054748000835577442*n^2+2550882268357014816*n-\ 141591568451397120)/n^2/(201111284400*n^10-4268454879000*n^9+40219013108100*n^8 -221501916311620*n^7+789529644942192*n^6-1903086863425990*n^5+3141658464550995* n^4-3507748068910267*n^3+2535599906207259*n^2-1071818629224903*n+ 201256810806114)/(3*n-1)^2/(3*n-2)^2*b(n-2)+4/81*(n-1)*(n-2)*(586440505310400*n ^14-13619695437784800*n^13+142231711903750800*n^12-882882552472392360*n^11+ 3628148568465988572*n^10-10405371442147524020*n^9+21393063306892695308*n^8-\ 31889587650802556632*n^7+34465531413211968071*n^6-26737365735758218336*n^5+ 14580360549067214051*n^4-5394625116788678624*n^3+1275871055962718202*n^2-\ 172435742329318272*n+10056781528118640)/n^2/(201111284400*n^10-4268454879000*n^ 9+40219013108100*n^8-221501916311620*n^7+789529644942192*n^6-1903086863425990*n ^5+3141658464550995*n^4-3507748068910267*n^3+2535599906207259*n^2-\ 1071818629224903*n+201256810806114)/(3*n-1)^2/(3*n-2)^2*b(n-3)-196/81*(n-1)*(n-\ 2)*(201111284400*n^10-2257342035000*n^9+10852926995100*n^8-29280832962820*n^7+ 48831757675652*n^6-52329773348458*n^5+36340723184225*n^4-16109138683447*n^3+ 4360283292188*n^2-651903197544*n+41018147280)*(n-3)^2*(2*n-7)^2/n^2/( 201111284400*n^10-4268454879000*n^9+40219013108100*n^8-221501916311620*n^7+ 789529644942192*n^6-1903086863425990*n^5+3141658464550995*n^4-3507748068910267* n^3+2535599906207259*n^2-1071818629224903*n+201256810806114)/(3*n-1)^2/(3*n-2)^ 2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 37, b(2) = 3925, b(3) = 524836 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 16 15 a(n) = 2/81 (128497239609398400 n - 3112761205396939200 n 14 13 + 34345417290576238800 n - 228803562403738788260 n 12 11 + 1028535770673700906092 n - 3302997028521842942488 n 10 9 + 7825272925840021644042 n - 13926235910289347242131 n 8 7 + 18777894159821846304942 n - 19207136041572044888704 n 6 5 + 14820174797314619699274 n - 8515242416561157904907 n 4 3 + 3562955073218115434190 n - 1047047813800287639030 n 2 + 203378003954398288260 n - 23292999112440123360 n + 1184133976143350400) / 2 2 2 a(n - 1) / (n %1 (3 n - 1) (3 n - 2) ) - 4/81 (n - 1) ( / 15 14 13 7089172775100000 n - 171730552810050000 n + 1887864001985318400 n 12 11 - 12471216714280649820 n + 55251258197116303740 n 10 9 - 173473154113028450800 n + 397642988067484830182 n 8 7 - 675451472451117045855 n + 854066396315000315160 n 6 5 - 800550325162481208888 n + 549260201123119756341 n 4 3 - 269646346685451368015 n + 91294144805520534301 n 2 - 20054748000835577442 n + 2550882268357014816 n - 141591568451397120) / 2 2 2 a(n - 2) / (n %1 (3 n - 1) (3 n - 2) ) + 4/81 (n - 1) (n - 2) ( / 14 13 12 586440505310400 n - 13619695437784800 n + 142231711903750800 n 11 10 - 882882552472392360 n + 3628148568465988572 n 9 8 - 10405371442147524020 n + 21393063306892695308 n 7 6 - 31889587650802556632 n + 34465531413211968071 n 5 4 - 26737365735758218336 n + 14580360549067214051 n 3 2 - 5394625116788678624 n + 1275871055962718202 n - 172435742329318272 n / 2 2 2 196 + 10056781528118640) a(n - 3) / (n %1 (3 n - 1) (3 n - 2) ) - --- / 81 10 9 8 (n - 1) (n - 2) (201111284400 n - 2257342035000 n + 10852926995100 n 7 6 5 - 29280832962820 n + 48831757675652 n - 52329773348458 n 4 3 2 + 36340723184225 n - 16109138683447 n + 4360283292188 n 2 2 / 2 - 651903197544 n + 41018147280) (n - 3) (2 n - 7) a(n - 4) / (n %1 / 2 2 (3 n - 1) (3 n - 2) ) 10 9 8 %1 := 201111284400 n - 4268454879000 n + 40219013108100 n 7 6 5 - 221501916311620 n + 789529644942192 n - 1903086863425990 n 4 3 2 + 3141658464550995 n - 3507748068910267 n + 2535599906207259 n - 1071818629224903 n + 201256810806114 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.02703717235871779214222271421111066382389065620124904781353367693549045402\ 703436274858368021974155964 The implied delta is, -0.4260205806131543151499842738728939964771979530565174\ 743665575856315344565440264349130221045576394809 Since this is negative, there is no Apery-style irrationality proof of, 0.027\ 037172358717792142222714211110663823890656201249047813533676935490454027\ 03436274858368021974155964, but still a very fast way to compute it to many digits ----------------------- This took, 5.710, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 ) binomial(n, k) binomial(2 n, k) / ----- k By Shalosh B. Ekhad Let ----- \ 2 b(n) = ) binomial(n, k) binomial(2 n, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 8 7 6 5 b(n) = 1/2 (3099984 n - 28140372 n + 105293798 n - 211459455 n 4 3 2 + 248705831 n - 175167927 n + 72209387 n - 15998526 n + 1481040) / 2 2 8 7 b(n - 1) / (n %1 (2 n - 1) ) - 1/4 (4817132 n - 48545063 n / 6 5 4 3 2 + 205423716 n - 474618440 n + 651106590 n - 539221337 n + 261162082 n / 2 2 - 67257000 n + 7128000) b(n - 2) / (n %1 (2 n - 1) ) + 32 (n - 1) / 4 3 2 2 (n - 2) (15196 n - 46767 n + 46967 n - 17766 n + 2376) (2 n - 5) / 2 2 b(n - 3) / (n %1 (2 n - 1) ) / 4 3 2 %1 := 15196 n - 107551 n + 278444 n - 312785 n + 129072 and in Maple notation b(n) = 1/2*(3099984*n^8-28140372*n^7+105293798*n^6-211459455*n^5+248705831*n^4-\ 175167927*n^3+72209387*n^2-15998526*n+1481040)/n^2/(15196*n^4-107551*n^3+278444 *n^2-312785*n+129072)/(2*n-1)^2*b(n-1)-1/4*(4817132*n^8-48545063*n^7+205423716* n^6-474618440*n^5+651106590*n^4-539221337*n^3+261162082*n^2-67257000*n+7128000) /n^2/(15196*n^4-107551*n^3+278444*n^2-312785*n+129072)/(2*n-1)^2*b(n-2)+32*(n-1 )*(n-2)*(15196*n^4-46767*n^3+46967*n^2-17766*n+2376)*(2*n-5)^2/n^2/(15196*n^4-\ 107551*n^3+278444*n^2-312785*n+129072)/(2*n-1)^2*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 5, b(2) = 69 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 8 7 6 5 a(n) = 1/2 (3099984 n - 28140372 n + 105293798 n - 211459455 n 4 3 2 + 248705831 n - 175167927 n + 72209387 n - 15998526 n + 1481040) / 2 2 8 7 a(n - 1) / (n %1 (2 n - 1) ) - 1/4 (4817132 n - 48545063 n / 6 5 4 3 2 + 205423716 n - 474618440 n + 651106590 n - 539221337 n + 261162082 n / 2 2 - 67257000 n + 7128000) a(n - 2) / (n %1 (2 n - 1) ) + 32 (n - 1) / 4 3 2 2 (n - 2) (15196 n - 46767 n + 46967 n - 17766 n + 2376) (2 n - 5) / 2 2 a(n - 3) / (n %1 (2 n - 1) ) / 4 3 2 %1 := 15196 n - 107551 n + 278444 n - 312785 n + 129072 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.11540096030846206360324718918385500969416401399135832589304692267386563200\ -2674 58557305472433586127661409 10 This constant is identified as, 0 The implied delta is, -0.4054297527601340017190668667680772358234424233226209\ 131288723103406022407555134394699094849546304178 Since this is negative, there is no Apery-style irrationality proof of, 0, but still a very fast way to compute it to many digits ----------------------- This took, 1.198, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(2 n, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(2 n, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 9 8 7 6 5 b(n) = (606112 n - 6696560 n + 31311040 n - 81017720 n + 127383010 n 4 3 2 - 125800019 n + 77834133 n - 29058120 n + 5952528 n - 516780) b(n - 1) / 2 2 9 8 7 / (n %1 (2 n - 1) ) - (2579200 n - 31075200 n + 160237512 n / 6 5 4 3 2 - 461710836 n + 813584418 n - 900625167 n + 618229715 n - 249438276 n / 2 2 + 52790004 n - 4579740) b(n - 2) / (n %1 (2 n - 1) ) + 243 / 5 4 3 2 2 (3224 n - 13052 n + 18966 n - 11917 n + 3013 n - 264) (n - 2) 2 / 2 2 (2 n - 5) b(n - 3) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 3224 n - 29172 n + 103414 n - 179367 n + 152073 n - 50436 and in Maple notation b(n) = (606112*n^9-6696560*n^8+31311040*n^7-81017720*n^6+127383010*n^5-\ 125800019*n^4+77834133*n^3-29058120*n^2+5952528*n-516780)/n^2/(3224*n^5-29172*n ^4+103414*n^3-179367*n^2+152073*n-50436)/(2*n-1)^2*b(n-1)-(2579200*n^9-31075200 *n^8+160237512*n^7-461710836*n^6+813584418*n^5-900625167*n^4+618229715*n^3-\ 249438276*n^2+52790004*n-4579740)/n^2/(3224*n^5-29172*n^4+103414*n^3-179367*n^2 +152073*n-50436)/(2*n-1)^2*b(n-2)+243*(3224*n^5-13052*n^4+18966*n^3-11917*n^2+ 3013*n-264)*(n-2)^2*(2*n-5)^2/n^2/(3224*n^5-29172*n^4+103414*n^3-179367*n^2+ 152073*n-50436)/(2*n-1)^2*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 9, b(2) = 209 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 9 8 7 6 5 a(n) = (606112 n - 6696560 n + 31311040 n - 81017720 n + 127383010 n 4 3 2 - 125800019 n + 77834133 n - 29058120 n + 5952528 n - 516780) a(n - 1) / 2 2 9 8 7 / (n %1 (2 n - 1) ) - (2579200 n - 31075200 n + 160237512 n / 6 5 4 3 2 - 461710836 n + 813584418 n - 900625167 n + 618229715 n - 249438276 n / 2 2 + 52790004 n - 4579740) a(n - 2) / (n %1 (2 n - 1) ) + 243 / 5 4 3 2 2 (3224 n - 13052 n + 18966 n - 11917 n + 3013 n - 264) (n - 2) 2 / 2 2 (2 n - 5) a(n - 3) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 3224 n - 29172 n + 103414 n - 179367 n + 152073 n - 50436 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.11552453009332421823620535357636276134591668906004254235344666824889893699\ 49491192676438878327364479 This constant is identified as, 1/6 ln(2) The implied delta is, -0.3907898617734913584606055080310586170624746174855498\ 315659960242935033874674844525990847233534063068 Since this is negative, there is no Apery-style irrationality proof of, 1/6 ln(2), but still a very fast way to compute it to many digits ----------------------- This took, 1.142, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(2 n, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(2 n, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 9 8 7 6 b(n) = 1/2 (10763280 n - 118640700 n + 558599778 n - 1467263289 n 5 4 3 2 + 2356991829 n - 2388280817 n + 1519606881 n - 584378578 n / 2 2 + 123559200 n - 11088000) b(n - 1) / (n %1 (2 n - 1) ) - 1/4 ( / 9 8 7 6 159236748 n - 1914459993 n + 9920682144 n - 28979437236 n 5 4 3 2 + 52364681190 n - 60390967259 n + 44201152958 n - 19698012616 n / 2 2 + 4842204544 n - 502740480) b(n - 2) / (n %1 (2 n - 1) ) + 1024 / 5 4 3 2 2 (19932 n - 80181 n + 125268 n - 94841 n + 35338 n - 5016) (n - 2) 2 / 2 2 (2 n - 5) b(n - 3) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 19932 n - 179841 n + 645312 n - 1151051 n + 1021208 n - 360576 and in Maple notation b(n) = 1/2*(10763280*n^9-118640700*n^8+558599778*n^7-1467263289*n^6+2356991829* n^5-2388280817*n^4+1519606881*n^3-584378578*n^2+123559200*n-11088000)/n^2/( 19932*n^5-179841*n^4+645312*n^3-1151051*n^2+1021208*n-360576)/(2*n-1)^2*b(n-1)-\ 1/4*(159236748*n^9-1914459993*n^8+9920682144*n^7-28979437236*n^6+52364681190*n^ 5-60390967259*n^4+44201152958*n^3-19698012616*n^2+4842204544*n-502740480)/n^2/( 19932*n^5-179841*n^4+645312*n^3-1151051*n^2+1021208*n-360576)/(2*n-1)^2*b(n-2)+ 1024*(19932*n^5-80181*n^4+125268*n^3-94841*n^2+35338*n-5016)*(n-2)^2*(2*n-5)^2/ n^2/(19932*n^5-179841*n^4+645312*n^3-1151051*n^2+1021208*n-360576)/(2*n-1)^2*b( n-3) Of course, the initial conditions are b(0) = 1, b(1) = 13, b(2) = 421 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 9 8 7 6 a(n) = 1/2 (10763280 n - 118640700 n + 558599778 n - 1467263289 n 5 4 3 2 + 2356991829 n - 2388280817 n + 1519606881 n - 584378578 n / 2 2 + 123559200 n - 11088000) a(n - 1) / (n %1 (2 n - 1) ) - 1/4 ( / 9 8 7 6 159236748 n - 1914459993 n + 9920682144 n - 28979437236 n 5 4 3 2 + 52364681190 n - 60390967259 n + 44201152958 n - 19698012616 n / 2 2 + 4842204544 n - 502740480) a(n - 2) / (n %1 (2 n - 1) ) + 1024 / 5 4 3 2 2 (19932 n - 80181 n + 125268 n - 94841 n + 35338 n - 5016) (n - 2) 2 / 2 2 (2 n - 5) a(n - 3) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 19932 n - 179841 n + 645312 n - 1151051 n + 1021208 n - 360576 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.08788898309344877531161961895380205637179924462581995613877554669099954345\ 748871734988926038509856710 This constant is identified as, 2/25 ln(3) The implied delta is, -0.5610958200446568429305585854717801795560060032163444\ 613374448448521986104062890257196862520357966507 Since this is negative, there is no Apery-style irrationality proof of, 2/25 ln(3), but still a very fast way to compute it to many digits ----------------------- This took, 1.421, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ ) binomial(n, k) binomial(2 n, k) binomial(2 n + k, k) / ----- k By Shalosh B. Ekhad Let ----- \ b(n) = ) binomial(n, k) binomial(2 n, k) binomial(2 n + k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 6 5 4 3 2 b(n) = (6528 n - 30192 n + 55040 n - 50406 n + 24465 n - 5985 n + 585) / 2 2 2 b(n - 1) / (n (48 n - 126 n + 83) (2 n - 1) ) / 2 2 2 (48 n - 30 n + 5) (n - 1) (2 n - 3) b(n - 2) - ----------------------------------------------- 2 2 2 n (48 n - 126 n + 83) (2 n - 1) and in Maple notation b(n) = (6528*n^6-30192*n^5+55040*n^4-50406*n^3+24465*n^2-5985*n+585)/n^2/(48*n^ 2-126*n+83)/(2*n-1)^2*b(n-1)-(48*n^2-30*n+5)*(n-1)^2*(2*n-3)^2/n^2/(48*n^2-126* n+83)/(2*n-1)^2*b(n-2) Of course, the initial conditions are b(0) = 1, b(1) = 7 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 6 5 4 3 2 a(n) = (6528 n - 30192 n + 55040 n - 50406 n + 24465 n - 5985 n + 585) / 2 2 2 a(n - 1) / (n (48 n - 126 n + 83) (2 n - 1) ) / 2 2 2 (48 n - 30 n + 5) (n - 1) (2 n - 3) a(n - 2) - ----------------------------------------------- 2 2 2 n (48 n - 126 n + 83) (2 n - 1) but with the following simpler initial conditions a(0) = 0, a(1) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.14303774494332403795412305796921958167121303488754769023787462864087021481\ 76696412029242522278051048 2 Pi This constant is identified as, --- 69 The implied delta is, -0.0604626290026458773344125018774411738927566375995041\ 179493927032480636635250768256088955552559271054 2 Pi Since this is negative, there is no Apery-style irrationality proof of, ---, 69 but still a very fast way to compute it to many digits ----------------------- This took, 3.863, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(2 n, k) binomial(2 n + k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(2 n, k) binomial(2 n + k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 9 8 7 6 5 b(n) = (1251264 n - 13674528 n + 63396928 n - 162926864 n + 254659060 n 4 3 2 - 250004822 n + 153635704 n - 56901029 n + 11542632 n - 985950) / 2 2 9 8 7 b(n - 1) / (n %1 (2 n - 1) ) + (6344128 n - 75676384 n + 386498560 n / 6 5 4 3 - 1103148176 n + 1925222884 n - 2110111922 n + 1433966688 n 2 / 2 2 - 573324535 n + 120654852 n - 10381950) b(n - 2) / (n %1 (2 n - 1) ) / 5 4 3 2 2 - (5488 n - 21560 n + 31052 n - 19954 n + 5604 n - 585) (n - 2) 2 / 2 2 (2 n - 5) b(n - 3) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 5488 n - 49000 n + 172172 n - 297350 n + 252348 n - 84243 and in Maple notation b(n) = (1251264*n^9-13674528*n^8+63396928*n^7-162926864*n^6+254659060*n^5-\ 250004822*n^4+153635704*n^3-56901029*n^2+11542632*n-985950)/n^2/(5488*n^5-49000 *n^4+172172*n^3-297350*n^2+252348*n-84243)/(2*n-1)^2*b(n-1)+(6344128*n^9-\ 75676384*n^8+386498560*n^7-1103148176*n^6+1925222884*n^5-2110111922*n^4+ 1433966688*n^3-573324535*n^2+120654852*n-10381950)/n^2/(5488*n^5-49000*n^4+ 172172*n^3-297350*n^2+252348*n-84243)/(2*n-1)^2*b(n-2)-(5488*n^5-21560*n^4+ 31052*n^3-19954*n^2+5604*n-585)*(n-2)^2*(2*n-5)^2/n^2/(5488*n^5-49000*n^4+ 172172*n^3-297350*n^2+252348*n-84243)/(2*n-1)^2*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 13, b(2) = 441 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 9 8 7 6 5 a(n) = (1251264 n - 13674528 n + 63396928 n - 162926864 n + 254659060 n 4 3 2 - 250004822 n + 153635704 n - 56901029 n + 11542632 n - 985950) / 2 2 9 8 7 a(n - 1) / (n %1 (2 n - 1) ) + (6344128 n - 75676384 n + 386498560 n / 6 5 4 3 - 1103148176 n + 1925222884 n - 2110111922 n + 1433966688 n 2 / 2 2 - 573324535 n + 120654852 n - 10381950) a(n - 2) / (n %1 (2 n - 1) ) / 5 4 3 2 2 - (5488 n - 21560 n + 31052 n - 19954 n + 5604 n - 585) (n - 2) 2 / 2 2 (2 n - 5) a(n - 3) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 5488 n - 49000 n + 172172 n - 297350 n + 252348 n - 84243 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.06931471805599453094172321214581765680755001343602552541206800094933936219\ 696947156058633269964186875 This constant is identified as, 1/10 ln(2) The implied delta is, -0.5774941402332961270519299428725279698884689265527361\ 110795077025862981700034163965916252497829383424 Since this is negative, there is no Apery-style irrationality proof of, 1/10 ln(2), but still a very fast way to compute it to many digits ----------------------- This took, 1.367, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(2 n, k) binomial(2 n + k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(2 n, k) binomial(2 n + k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 9 8 7 6 b(n) = (11556864 n - 125528832 n + 578172192 n - 1475901768 n 5 4 3 2 + 2291922084 n - 2237526190 n + 1370127087 n - 507379955 n / 2 2 9 + 103388163 n - 8896050) b(n - 1) / (n %1 (2 n - 1) ) + (243175680 n / 8 7 6 5 - 2884511520 n + 14627584464 n - 41370656304 n + 71342328120 n 4 3 2 - 76949350618 n + 51145289521 n - 19812569744 n + 3985645235 n / 2 2 - 324939750) b(n - 2) / (n %1 (2 n - 1) ) - 2 / 5 4 3 2 2 (40128 n - 154968 n + 217692 n - 134698 n + 35731 n - 3495) (n - 2) 2 / 2 2 (2 n - 5) b(n - 3) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 40128 n - 355608 n + 1238844 n - 2118862 n + 1778715 n - 586712 and in Maple notation b(n) = (11556864*n^9-125528832*n^8+578172192*n^7-1475901768*n^6+2291922084*n^5-\ 2237526190*n^4+1370127087*n^3-507379955*n^2+103388163*n-8896050)/n^2/(40128*n^5 -355608*n^4+1238844*n^3-2118862*n^2+1778715*n-586712)/(2*n-1)^2*b(n-1)+( 243175680*n^9-2884511520*n^8+14627584464*n^7-41370656304*n^6+71342328120*n^5-\ 76949350618*n^4+51145289521*n^3-19812569744*n^2+3985645235*n-324939750)/n^2/( 40128*n^5-355608*n^4+1238844*n^3-2118862*n^2+1778715*n-586712)/(2*n-1)^2*b(n-2) -2*(40128*n^5-154968*n^4+217692*n^3-134698*n^2+35731*n-3495)*(n-2)^2*(2*n-5)^2/ n^2/(40128*n^5-355608*n^4+1238844*n^3-2118862*n^2+1778715*n-586712)/(2*n-1)^2*b (n-3) Of course, the initial conditions are b(0) = 1, b(1) = 19, b(2) = 931 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 9 8 7 6 a(n) = (11556864 n - 125528832 n + 578172192 n - 1475901768 n 5 4 3 2 + 2291922084 n - 2237526190 n + 1370127087 n - 507379955 n / 2 2 9 + 103388163 n - 8896050) a(n - 1) / (n %1 (2 n - 1) ) + (243175680 n / 8 7 6 5 - 2884511520 n + 14627584464 n - 41370656304 n + 71342328120 n 4 3 2 - 76949350618 n + 51145289521 n - 19812569744 n + 3985645235 n / 2 2 - 324939750) a(n - 2) / (n %1 (2 n - 1) ) - 2 / 5 4 3 2 2 (40128 n - 154968 n + 217692 n - 134698 n + 35731 n - 3495) (n - 2) 2 / 2 2 (2 n - 5) a(n - 3) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 40128 n - 355608 n + 1238844 n - 2118862 n + 1778715 n - 586712 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.04225431879492729582289404757394329633259579068549036352825747437067285743\ 148496026436983672360508034 This constant is identified as, 1/26 ln(3) The implied delta is, -0.7437866765777699020226367843238357979070319837434721\ 161166140012974924355327685688630091692370339409 Since this is negative, there is no Apery-style irrationality proof of, 1/26 ln(3), but still a very fast way to compute it to many digits ----------------------- This took, 1.594, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ ) binomial(n, k) binomial(2 n, k) binomial(2 n + 2 k, k) / ----- k By Shalosh B. Ekhad Let ----- \ b(n) = ) binomial(n, k) binomial(2 n, k) binomial(2 n + 2 k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 22 21 b(n) = 1/12 (1639513016779632768 n - 53910416451510119232 n 20 19 + 830072450788236868992 n - 7956851638960649880160 n 18 17 + 53252041472084137404456 n - 264499500376448891889892 n 16 15 + 1011741506944383210720764 n - 3052469750624653163486632 n 14 13 + 7379931977041839436468616 n - 14447065971290520540737567 n 12 11 + 23045427368838340517524412 n - 30047624432061028761258221 n 10 9 + 32030823942973132138119268 n - 27848322173621377316531387 n 8 7 + 19643658815401034030830276 n - 11147248503722087490065967 n 6 5 + 5026193678411479875215572 n - 1768927863123393877442022 n 4 3 + 473603880303164077794156 n - 92809014800222694163320 n 2 + 12506052865184969364720 n - 1032238090527948561600 n / 2 + 39233793865875840000) b(n - 1) / (n (4 n - 3) (3 n - 1) (3 n - 2) / 2 21 (4 n - 1) %1 (2 n - 1) ) + 1/2 (2 n - 3) (670693534593699456 n 20 19 - 21718377351528834816 n + 328596081210820234368 n 18 17 - 3087193851575859029664 n + 20189524303044247378296 n 16 15 - 97642750906935044479560 n + 362149002268971676100480 n 14 13 - 1054165113426369325409456 n + 2444399523838901180103332 n 12 11 - 4557001780129530541567722 n + 6863724776836985779769611 n 10 9 - 8363519788943343941603410 n + 8228568660966128691821879 n 8 7 - 6502896685926921575181112 n + 4091957495643413813726221 n 6 5 - 2023708901766177331071906 n + 772168708486260366718577 n 4 3 - 221389000949527146674090 n + 45872596076056070518820 n 2 - 6455684199018518100264 n + 550519917163258728960 n / 2 - 21456613457693568000) b(n - 2) / (n (4 n - 3) (3 n - 1) (3 n - 2) / 2 18 (4 n - 1) %1 (2 n - 1) ) + (2 n - 3) (2 n - 5) (10997332632390528 n 17 16 - 295629956641658304 n + 3650771351387965248 n 15 14 - 27481211957287202112 n + 141099108420654830184 n 13 12 - 523928388876001820308 n + 1455685020034093887228 n 11 10 - 3088845481019532568372 n + 5064726784830846944946 n 9 8 - 6451907413647681669307 n + 6386544591741674491950 n 7 6 - 4889355966791524986830 n + 2866706330281325648064 n 5 4 - 1266924355365713970311 n + 411978257627804606334 n 3 2 - 95008211263362452396 n + 14635676564635537038 n 2 / 2 - 1346881942368923580 n + 56002064767110000) (n - 2) b(n - 3) / (n / 2 (4 n - 3) (3 n - 1) (3 n - 2) (4 n - 1) %1 (2 n - 1) ) + 9 (2 n - 3) 14 13 12 (2 n - 5) (5989832588448 n - 89140503841488 n + 595774252740648 n 11 10 9 - 2367022104074924 n + 6235750221135948 n - 11502479919483460 n 8 7 6 + 15289662094565532 n - 14845167670943811 n + 10555194555981042 n 5 4 3 - 5457214243694632 n + 2016044165229840 n - 515640016794845 n 2 2 2 + 86386787952222 n - 8511914500920 n + 374445126000) (2 n - 7) (n - 2) 2 / 2 (n - 3) b(n - 4) / (n (4 n - 3) (3 n - 1) (3 n - 2) (4 n - 1) %1 / 2 (2 n - 1) ) 14 13 12 %1 := 5989832588448 n - 172998160079760 n + 2299675568228760 n 11 10 9 - 18649571498793836 n + 103084100566544896 n - 410843638546643136 n 8 7 6 + 1217598939307121952 n - 2726034480616051467 n + 4633309960045658727 n 5 4 3 - 5949612399085521619 n + 5682064097238550343 n - 3913859601722074602 n 2 + 1838192079866743314 n - 526952131079799780 n + 69570352728653760 and in Maple notation b(n) = 1/12*(1639513016779632768*n^22-53910416451510119232*n^21+ 830072450788236868992*n^20-7956851638960649880160*n^19+53252041472084137404456* n^18-264499500376448891889892*n^17+1011741506944383210720764*n^16-\ 3052469750624653163486632*n^15+7379931977041839436468616*n^14-\ 14447065971290520540737567*n^13+23045427368838340517524412*n^12-\ 30047624432061028761258221*n^11+32030823942973132138119268*n^10-\ 27848322173621377316531387*n^9+19643658815401034030830276*n^8-\ 11147248503722087490065967*n^7+5026193678411479875215572*n^6-\ 1768927863123393877442022*n^5+473603880303164077794156*n^4-\ 92809014800222694163320*n^3+12506052865184969364720*n^2-1032238090527948561600* n+39233793865875840000)/n^2/(4*n-3)/(3*n-1)/(3*n-2)/(4*n-1)/(5989832588448*n^14 -172998160079760*n^13+2299675568228760*n^12-18649571498793836*n^11+ 103084100566544896*n^10-410843638546643136*n^9+1217598939307121952*n^8-\ 2726034480616051467*n^7+4633309960045658727*n^6-5949612399085521619*n^5+ 5682064097238550343*n^4-3913859601722074602*n^3+1838192079866743314*n^2-\ 526952131079799780*n+69570352728653760)/(2*n-1)^2*b(n-1)+1/2*(2*n-3)*( 670693534593699456*n^21-21718377351528834816*n^20+328596081210820234368*n^19-\ 3087193851575859029664*n^18+20189524303044247378296*n^17-\ 97642750906935044479560*n^16+362149002268971676100480*n^15-\ 1054165113426369325409456*n^14+2444399523838901180103332*n^13-\ 4557001780129530541567722*n^12+6863724776836985779769611*n^11-\ 8363519788943343941603410*n^10+8228568660966128691821879*n^9-\ 6502896685926921575181112*n^8+4091957495643413813726221*n^7-\ 2023708901766177331071906*n^6+772168708486260366718577*n^5-\ 221389000949527146674090*n^4+45872596076056070518820*n^3-6455684199018518100264 *n^2+550519917163258728960*n-21456613457693568000)/n^2/(4*n-3)/(3*n-1)/(3*n-2)/ (4*n-1)/(5989832588448*n^14-172998160079760*n^13+2299675568228760*n^12-\ 18649571498793836*n^11+103084100566544896*n^10-410843638546643136*n^9+ 1217598939307121952*n^8-2726034480616051467*n^7+4633309960045658727*n^6-\ 5949612399085521619*n^5+5682064097238550343*n^4-3913859601722074602*n^3+ 1838192079866743314*n^2-526952131079799780*n+69570352728653760)/(2*n-1)^2*b(n-2 )+(2*n-3)*(2*n-5)*(10997332632390528*n^18-295629956641658304*n^17+ 3650771351387965248*n^16-27481211957287202112*n^15+141099108420654830184*n^14-\ 523928388876001820308*n^13+1455685020034093887228*n^12-3088845481019532568372*n ^11+5064726784830846944946*n^10-6451907413647681669307*n^9+ 6386544591741674491950*n^8-4889355966791524986830*n^7+2866706330281325648064*n^ 6-1266924355365713970311*n^5+411978257627804606334*n^4-95008211263362452396*n^3 +14635676564635537038*n^2-1346881942368923580*n+56002064767110000)*(n-2)^2/n^2/ (4*n-3)/(3*n-1)/(3*n-2)/(4*n-1)/(5989832588448*n^14-172998160079760*n^13+ 2299675568228760*n^12-18649571498793836*n^11+103084100566544896*n^10-\ 410843638546643136*n^9+1217598939307121952*n^8-2726034480616051467*n^7+ 4633309960045658727*n^6-5949612399085521619*n^5+5682064097238550343*n^4-\ 3913859601722074602*n^3+1838192079866743314*n^2-526952131079799780*n+ 69570352728653760)/(2*n-1)^2*b(n-3)+9*(2*n-3)*(2*n-5)*(5989832588448*n^14-\ 89140503841488*n^13+595774252740648*n^12-2367022104074924*n^11+6235750221135948 *n^10-11502479919483460*n^9+15289662094565532*n^8-14845167670943811*n^7+ 10555194555981042*n^6-5457214243694632*n^5+2016044165229840*n^4-515640016794845 *n^3+86386787952222*n^2-8511914500920*n+374445126000)*(2*n-7)^2*(n-2)^2*(n-3)^2 /n^2/(4*n-3)/(3*n-1)/(3*n-2)/(4*n-1)/(5989832588448*n^14-172998160079760*n^13+ 2299675568228760*n^12-18649571498793836*n^11+103084100566544896*n^10-\ 410843638546643136*n^9+1217598939307121952*n^8-2726034480616051467*n^7+ 4633309960045658727*n^6-5949612399085521619*n^5+5682064097238550343*n^4-\ 3913859601722074602*n^3+1838192079866743314*n^2-526952131079799780*n+ 69570352728653760)/(2*n-1)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 9, b(2) = 217, b(3) = 6570 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 22 21 a(n) = 1/12 (1639513016779632768 n - 53910416451510119232 n 20 19 + 830072450788236868992 n - 7956851638960649880160 n 18 17 + 53252041472084137404456 n - 264499500376448891889892 n 16 15 + 1011741506944383210720764 n - 3052469750624653163486632 n 14 13 + 7379931977041839436468616 n - 14447065971290520540737567 n 12 11 + 23045427368838340517524412 n - 30047624432061028761258221 n 10 9 + 32030823942973132138119268 n - 27848322173621377316531387 n 8 7 + 19643658815401034030830276 n - 11147248503722087490065967 n 6 5 + 5026193678411479875215572 n - 1768927863123393877442022 n 4 3 + 473603880303164077794156 n - 92809014800222694163320 n 2 + 12506052865184969364720 n - 1032238090527948561600 n / 2 + 39233793865875840000) a(n - 1) / (n (4 n - 3) (3 n - 1) (3 n - 2) / 2 21 (4 n - 1) %1 (2 n - 1) ) + 1/2 (2 n - 3) (670693534593699456 n 20 19 - 21718377351528834816 n + 328596081210820234368 n 18 17 - 3087193851575859029664 n + 20189524303044247378296 n 16 15 - 97642750906935044479560 n + 362149002268971676100480 n 14 13 - 1054165113426369325409456 n + 2444399523838901180103332 n 12 11 - 4557001780129530541567722 n + 6863724776836985779769611 n 10 9 - 8363519788943343941603410 n + 8228568660966128691821879 n 8 7 - 6502896685926921575181112 n + 4091957495643413813726221 n 6 5 - 2023708901766177331071906 n + 772168708486260366718577 n 4 3 - 221389000949527146674090 n + 45872596076056070518820 n 2 - 6455684199018518100264 n + 550519917163258728960 n / 2 - 21456613457693568000) a(n - 2) / (n (4 n - 3) (3 n - 1) (3 n - 2) / 2 18 (4 n - 1) %1 (2 n - 1) ) + (2 n - 3) (2 n - 5) (10997332632390528 n 17 16 - 295629956641658304 n + 3650771351387965248 n 15 14 - 27481211957287202112 n + 141099108420654830184 n 13 12 - 523928388876001820308 n + 1455685020034093887228 n 11 10 - 3088845481019532568372 n + 5064726784830846944946 n 9 8 - 6451907413647681669307 n + 6386544591741674491950 n 7 6 - 4889355966791524986830 n + 2866706330281325648064 n 5 4 - 1266924355365713970311 n + 411978257627804606334 n 3 2 - 95008211263362452396 n + 14635676564635537038 n 2 / 2 - 1346881942368923580 n + 56002064767110000) (n - 2) a(n - 3) / (n / 2 (4 n - 3) (3 n - 1) (3 n - 2) (4 n - 1) %1 (2 n - 1) ) + 9 (2 n - 3) 14 13 12 (2 n - 5) (5989832588448 n - 89140503841488 n + 595774252740648 n 11 10 9 - 2367022104074924 n + 6235750221135948 n - 11502479919483460 n 8 7 6 + 15289662094565532 n - 14845167670943811 n + 10555194555981042 n 5 4 3 - 5457214243694632 n + 2016044165229840 n - 515640016794845 n 2 2 2 + 86386787952222 n - 8511914500920 n + 374445126000) (2 n - 7) (n - 2) 2 / 2 (n - 3) a(n - 4) / (n (4 n - 3) (3 n - 1) (3 n - 2) (4 n - 1) %1 / 2 (2 n - 1) ) 14 13 12 %1 := 5989832588448 n - 172998160079760 n + 2299675568228760 n 11 10 9 - 18649571498793836 n + 103084100566544896 n - 410843638546643136 n 8 7 6 + 1217598939307121952 n - 2726034480616051467 n + 4633309960045658727 n 5 4 3 - 5949612399085521619 n + 5682064097238550343 n - 3913859601722074602 n 2 + 1838192079866743314 n - 526952131079799780 n + 69570352728653760 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately -0.9321221664131375477791502027835640409617622759100764424449387320434703061\ -2009 262608102754617610852308973 10 This constant is identified as, 0 The implied delta is, -0.6587946227834138573668654075939522530992914386949486\ 941000623028593072881580365452888586212730339907 Since this is negative, there is no Apery-style irrationality proof of, 0, but still a very fast way to compute it to many digits ----------------------- This took, 2.280, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(2 n, k) binomial(2 n + 2 k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(2 n, k) binomial(2 n + 2 k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 22 21 b(n) = 1/4 (155369905048227151872 n - 5089158340315927805952 n 20 19 + 78046725411132595200000 n - 745074007490432632688640 n 18 17 + 4965709567303271473389312 n - 24560773572160705969205632 n 16 15 + 93555107887202914890911808 n - 281105341121720168563531712 n 14 13 + 676967941988222141230866496 n - 1320448870973526151851953576 n 12 11 + 2099635908686330234306643412 n - 2730586214348333186667776208 n 10 9 + 2905793907305417640087087835 n - 2524783384985815342920219829 n 8 7 + 1782307780154791671143980817 n - 1013935669217170141811079718 n 6 5 + 459251842829065219296860279 n - 162740169091882368342315791 n 4 3 + 43978405018027953967853513 n - 8719339826132035799954622 n 2 + 1191031351991560858921056 n - 99756439634181484062720 n / 2 + 3845880948066208128000) b(n - 1) / (n (4 n - 3) (3 n - 1) (3 n - 2) / 2 21 (4 n - 1) %1 (2 n - 1) ) + 1/4 (2 n - 3) (3449266344140169019392 n 20 19 - 111256465566473997336576 n + 1676059217865851067182592 n 18 17 - 15672271813909429516660224 n + 101957963852152238174548992 n 16 15 - 490249157085132848181310144 n + 1806605180538035892883579392 n 14 13 - 5221026153654765873961111792 n + 12009035379014833018727921264 n 12 11 - 22184754973639381030545086644 n + 33070753346157929862585538370 n 10 9 - 39824585143131640107134430263 n + 38655407914592598312682015697 n 8 7 - 30075352395800424201435045749 n + 18584959450485367795601560672 n 6 5 - 8998946459443513662256279439 n + 3349655484556500935160262615 n 4 3 - 932937904685841590090922537 n + 186898377430767768137881774 n 2 - 25308301417485061427675232 n + 2068615974888140183605440 n / 2 - 77203172328411308256000) b(n - 2) / (n (4 n - 3) (3 n - 1) (3 n - 2) / 2 (4 n - 1) %1 (2 n - 1) ) + 1/4 (2 n - 3) (2 n - 5) ( 18 17 60426671599266693120 n - 1616722249372675891200 n 16 15 + 19857872573948854262016 n - 148565640600510193233792 n 14 13 + 757468839206344398755520 n - 2790197388293016149196224 n 12 11 + 7681508128485838270083008 n - 16128687975193399469118344 n 10 9 + 26126741066443963788375900 n - 32818339666139939007428928 n 8 7 + 31959927731501632897816029 n - 24005790182460947465169889 n 6 5 + 13763938263739534687289203 n - 5925050682104017170122986 n 4 3 + 1868022407433883667580321 n - 415500151230473408174127 n 2 + 61408858026549265087563 n - 5399588808507071472390 n 2 / 2 + 214382165852085462000) (n - 2) b(n - 3) / (n (4 n - 3) (3 n - 1) / 2 343 (3 n - 2) (4 n - 1) %1 (2 n - 1) ) + --- (2 n - 3) (2 n - 5) ( 12 14 13 12 1260464572366848 n - 18598293495650304 n + 123085532660073024 n 11 10 - 483519442331890400 n + 1257304364926960224 n 9 8 7 - 2284574529366932728 n + 2984212691721284784 n - 2839104913685979864 n 6 5 4 + 1971119481111100077 n - 990929459254264747 n + 354188310910325322 n 3 2 - 87156871291560707 n + 13968781532813721 n - 1311073649114130 n 2 2 2 / 2 + 54930937050000) (2 n - 7) (n - 2) (n - 3) b(n - 4) / (n (4 n - 3) / 2 (3 n - 1) (3 n - 2) (4 n - 1) %1 (2 n - 1) ) 14 13 12 %1 := 1260464572366848 n - 36244797508786176 n + 479565624188910144 n 11 10 - 3870021831255023072 n + 21280500362837776000 n 9 8 - 84351234615374969304 n + 248553305368954538088 n 7 6 - 553119757183101674232 n + 934157847631187334453 n 5 4 - 1191581028958437461185 n + 1130077755080831712188 n 3 2 - 772738407517231831077 n + 360161777757167834247 n - 102425651597341043802 n + 13410389141447366880 and in Maple notation b(n) = 1/4*(155369905048227151872*n^22-5089158340315927805952*n^21+ 78046725411132595200000*n^20-745074007490432632688640*n^19+ 4965709567303271473389312*n^18-24560773572160705969205632*n^17+ 93555107887202914890911808*n^16-281105341121720168563531712*n^15+ 676967941988222141230866496*n^14-1320448870973526151851953576*n^13+ 2099635908686330234306643412*n^12-2730586214348333186667776208*n^11+ 2905793907305417640087087835*n^10-2524783384985815342920219829*n^9+ 1782307780154791671143980817*n^8-1013935669217170141811079718*n^7+ 459251842829065219296860279*n^6-162740169091882368342315791*n^5+ 43978405018027953967853513*n^4-8719339826132035799954622*n^3+ 1191031351991560858921056*n^2-99756439634181484062720*n+3845880948066208128000) /n^2/(4*n-3)/(3*n-1)/(3*n-2)/(4*n-1)/(1260464572366848*n^14-36244797508786176*n ^13+479565624188910144*n^12-3870021831255023072*n^11+21280500362837776000*n^10-\ 84351234615374969304*n^9+248553305368954538088*n^8-553119757183101674232*n^7+ 934157847631187334453*n^6-1191581028958437461185*n^5+1130077755080831712188*n^4 -772738407517231831077*n^3+360161777757167834247*n^2-102425651597341043802*n+ 13410389141447366880)/(2*n-1)^2*b(n-1)+1/4*(2*n-3)*(3449266344140169019392*n^21 -111256465566473997336576*n^20+1676059217865851067182592*n^19-\ 15672271813909429516660224*n^18+101957963852152238174548992*n^17-\ 490249157085132848181310144*n^16+1806605180538035892883579392*n^15-\ 5221026153654765873961111792*n^14+12009035379014833018727921264*n^13-\ 22184754973639381030545086644*n^12+33070753346157929862585538370*n^11-\ 39824585143131640107134430263*n^10+38655407914592598312682015697*n^9-\ 30075352395800424201435045749*n^8+18584959450485367795601560672*n^7-\ 8998946459443513662256279439*n^6+3349655484556500935160262615*n^5-\ 932937904685841590090922537*n^4+186898377430767768137881774*n^3-\ 25308301417485061427675232*n^2+2068615974888140183605440*n-\ 77203172328411308256000)/n^2/(4*n-3)/(3*n-1)/(3*n-2)/(4*n-1)/(1260464572366848* n^14-36244797508786176*n^13+479565624188910144*n^12-3870021831255023072*n^11+ 21280500362837776000*n^10-84351234615374969304*n^9+248553305368954538088*n^8-\ 553119757183101674232*n^7+934157847631187334453*n^6-1191581028958437461185*n^5+ 1130077755080831712188*n^4-772738407517231831077*n^3+360161777757167834247*n^2-\ 102425651597341043802*n+13410389141447366880)/(2*n-1)^2*b(n-2)+1/4*(2*n-3)*(2*n -5)*(60426671599266693120*n^18-1616722249372675891200*n^17+ 19857872573948854262016*n^16-148565640600510193233792*n^15+ 757468839206344398755520*n^14-2790197388293016149196224*n^13+ 7681508128485838270083008*n^12-16128687975193399469118344*n^11+ 26126741066443963788375900*n^10-32818339666139939007428928*n^9+ 31959927731501632897816029*n^8-24005790182460947465169889*n^7+ 13763938263739534687289203*n^6-5925050682104017170122986*n^5+ 1868022407433883667580321*n^4-415500151230473408174127*n^3+ 61408858026549265087563*n^2-5399588808507071472390*n+214382165852085462000)*(n-\ 2)^2/n^2/(4*n-3)/(3*n-1)/(3*n-2)/(4*n-1)/(1260464572366848*n^14-\ 36244797508786176*n^13+479565624188910144*n^12-3870021831255023072*n^11+ 21280500362837776000*n^10-84351234615374969304*n^9+248553305368954538088*n^8-\ 553119757183101674232*n^7+934157847631187334453*n^6-1191581028958437461185*n^5+ 1130077755080831712188*n^4-772738407517231831077*n^3+360161777757167834247*n^2-\ 102425651597341043802*n+13410389141447366880)/(2*n-1)^2*b(n-3)+343/12*(2*n-3)*( 2*n-5)*(1260464572366848*n^14-18598293495650304*n^13+123085532660073024*n^12-\ 483519442331890400*n^11+1257304364926960224*n^10-2284574529366932728*n^9+ 2984212691721284784*n^8-2839104913685979864*n^7+1971119481111100077*n^6-\ 990929459254264747*n^5+354188310910325322*n^4-87156871291560707*n^3+ 13968781532813721*n^2-1311073649114130*n+54930937050000)*(2*n-7)^2*(n-2)^2*(n-3 )^2/n^2/(4*n-3)/(3*n-1)/(3*n-2)/(4*n-1)/(1260464572366848*n^14-\ 36244797508786176*n^13+479565624188910144*n^12-3870021831255023072*n^11+ 21280500362837776000*n^10-84351234615374969304*n^9+248553305368954538088*n^8-\ 553119757183101674232*n^7+934157847631187334453*n^6-1191581028958437461185*n^5+ 1130077755080831712188*n^4-772738407517231831077*n^3+360161777757167834247*n^2-\ 102425651597341043802*n+13410389141447366880)/(2*n-1)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 17, b(2) = 769, b(3) = 43589 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 22 21 a(n) = 1/4 (155369905048227151872 n - 5089158340315927805952 n 20 19 + 78046725411132595200000 n - 745074007490432632688640 n 18 17 + 4965709567303271473389312 n - 24560773572160705969205632 n 16 15 + 93555107887202914890911808 n - 281105341121720168563531712 n 14 13 + 676967941988222141230866496 n - 1320448870973526151851953576 n 12 11 + 2099635908686330234306643412 n - 2730586214348333186667776208 n 10 9 + 2905793907305417640087087835 n - 2524783384985815342920219829 n 8 7 + 1782307780154791671143980817 n - 1013935669217170141811079718 n 6 5 + 459251842829065219296860279 n - 162740169091882368342315791 n 4 3 + 43978405018027953967853513 n - 8719339826132035799954622 n 2 + 1191031351991560858921056 n - 99756439634181484062720 n / 2 + 3845880948066208128000) a(n - 1) / (n (4 n - 3) (3 n - 1) (3 n - 2) / 2 21 (4 n - 1) %1 (2 n - 1) ) + 1/4 (2 n - 3) (3449266344140169019392 n 20 19 - 111256465566473997336576 n + 1676059217865851067182592 n 18 17 - 15672271813909429516660224 n + 101957963852152238174548992 n 16 15 - 490249157085132848181310144 n + 1806605180538035892883579392 n 14 13 - 5221026153654765873961111792 n + 12009035379014833018727921264 n 12 11 - 22184754973639381030545086644 n + 33070753346157929862585538370 n 10 9 - 39824585143131640107134430263 n + 38655407914592598312682015697 n 8 7 - 30075352395800424201435045749 n + 18584959450485367795601560672 n 6 5 - 8998946459443513662256279439 n + 3349655484556500935160262615 n 4 3 - 932937904685841590090922537 n + 186898377430767768137881774 n 2 - 25308301417485061427675232 n + 2068615974888140183605440 n / 2 - 77203172328411308256000) a(n - 2) / (n (4 n - 3) (3 n - 1) (3 n - 2) / 2 (4 n - 1) %1 (2 n - 1) ) + 1/4 (2 n - 3) (2 n - 5) ( 18 17 60426671599266693120 n - 1616722249372675891200 n 16 15 + 19857872573948854262016 n - 148565640600510193233792 n 14 13 + 757468839206344398755520 n - 2790197388293016149196224 n 12 11 + 7681508128485838270083008 n - 16128687975193399469118344 n 10 9 + 26126741066443963788375900 n - 32818339666139939007428928 n 8 7 + 31959927731501632897816029 n - 24005790182460947465169889 n 6 5 + 13763938263739534687289203 n - 5925050682104017170122986 n 4 3 + 1868022407433883667580321 n - 415500151230473408174127 n 2 + 61408858026549265087563 n - 5399588808507071472390 n 2 / 2 + 214382165852085462000) (n - 2) a(n - 3) / (n (4 n - 3) (3 n - 1) / 2 343 (3 n - 2) (4 n - 1) %1 (2 n - 1) ) + --- (2 n - 3) (2 n - 5) ( 12 14 13 12 1260464572366848 n - 18598293495650304 n + 123085532660073024 n 11 10 - 483519442331890400 n + 1257304364926960224 n 9 8 7 - 2284574529366932728 n + 2984212691721284784 n - 2839104913685979864 n 6 5 4 + 1971119481111100077 n - 990929459254264747 n + 354188310910325322 n 3 2 - 87156871291560707 n + 13968781532813721 n - 1311073649114130 n 2 2 2 / 2 + 54930937050000) (2 n - 7) (n - 2) (n - 3) a(n - 4) / (n (4 n - 3) / 2 (3 n - 1) (3 n - 2) (4 n - 1) %1 (2 n - 1) ) 14 13 12 %1 := 1260464572366848 n - 36244797508786176 n + 479565624188910144 n 11 10 - 3870021831255023072 n + 21280500362837776000 n 9 8 - 84351234615374969304 n + 248553305368954538088 n 7 6 - 553119757183101674232 n + 934157847631187334453 n 5 4 - 1191581028958437461185 n + 1130077755080831712188 n 3 2 - 772738407517231831077 n + 360161777757167834247 n - 102425651597341043802 n + 13410389141447366880 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.03353937970451348271373703813507305974558871617872202842519419400774485267\ 595297010996112872563316230 This constant is identified as, 3/62 ln(2) The implied delta is, -0.8579489227885534049398032813798596531368463658904285\ 475186717030499888912873791039845279316236161001 Since this is negative, there is no Apery-style irrationality proof of, 3/62 ln(2), but still a very fast way to compute it to many digits ----------------------- This took, 2.468, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(2 n, k) binomial(2 n + 2 k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(2 n, k) binomial(2 n + 2 k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 22 21 b(n) = 1/108 (1773202522138592223040128 n - 57968383229470970097940800 n 20 19 + 887350263343019533619913216 n - 8456888142633929237962932064 n 18 17 + 56283876727733765752749935112 n - 278109560393144559014410568836 n 16 + 1058932888815707720474433818900 n 15 - 3183100668992380937441875509256 n 14 + 7677255208641113541703728593744 n 13 - 15019140486763404798373859483003 n 12 + 23997613092764794651370342959556 n 11 - 31435161897450627453521248241825 n 10 + 33794565042858297182576753524924 n 9 - 29770539630405502463187380040071 n 8 + 21397098018899382044554800194164 n 7 6 - 12452457741046333450562147983371 n + 5799305179305846117873035976052 n 5 4 - 2123726263111614035468525389710 n + 595779869185627487035029176124 n 3 2 - 123031857893879428466892291864 n + 17529038371951857657287020080 n - 1529540763531984398189131200 n + 61164659259102399960960000) b(n - 1) / 2 2 / (n (4 n - 3) (3 n - 1) (3 n - 2) (4 n - 1) %1 (2 n - 1) ) + 1/18 / 21 20 (2 n - 3) (30259698266906326074819456 n - 974100520011648253216591872 n 19 18 + 14642526759952397444831748480 n - 136584303069243712851959905248 n 17 16 + 886155394517793048834614455656 n - 4247989178691186268353778021768 n 15 + 15600598894277331233878611851104 n 14 - 44910623758824651837181891313800 n 13 + 102844961970576342040207311272068 n 12 - 189030088283080695815622606640938 n 11 + 280144601672964417743740822768799 n 10 - 335070301686498614803334562958530 n 9 + 322647379911762224561098019583115 n 8 - 248668638218303614924866255820688 n 7 + 151934618964420348980407511839097 n 6 - 72568166810799071926515478141482 n 5 4 + 26564860267833594568999432043005 n - 7248651801037048460687141137634 n 3 2 + 1415968703080770455911648534420 n - 185927795943276484548438053640 n + 14652571406718771356195577600 n - 525191635382693403358080000) b(n - 2) / 2 2 / (n (4 n - 3) (3 n - 1) (3 n - 2) (4 n - 1) %1 (2 n - 1) ) + 1/9 / 18 (2 n - 3) (2 n - 5) (180101564931928329376128 n 17 16 - 4807153799245961441284032 n + 58883933773094347527314880 n 15 14 - 439152860714772227893817984 n + 2230927533690048272526574888 n 13 12 - 8183331164817549768520253252 n + 22419130077139891605177862156 n 11 10 - 46804703028238380346930764828 n + 75310626117753087284698375058 n 9 8 - 93848684664415700238198661727 n + 90528074034717438503367828518 n 7 6 - 67221177324360101270248707070 n + 38006234848095368197742784584 n 5 4 - 16081404118748913572910517611 n + 4962915567782881865188321318 n 3 2 - 1074969937225421035752299660 n + 153767358522183936619842294 n 2 - 13003982416281808791728460 n + 494581794236591351598000) (n - 2) / 2 2 b(n - 3) / (n (4 n - 3) (3 n - 1) (3 n - 2) (4 n - 1) %1 (2 n - 1) ) + / 1331 14 13 ---- (2 n - 3) (2 n - 5) (639029665947318048 n - 9388208107065177936 n 27 12 11 + 61818971391460118568 n - 241412143113034925932 n 10 9 + 623394337356699491844 n - 1123435837985043239228 n 8 7 + 1453107080687995223652 n - 1366126413770221539495 n 6 5 + 934817253445320523170 n - 461619297950334419264 n 4 3 + 161357471792728350120 n - 38614124995148396137 n 2 + 5978770560190065750 n - 538466543363273976 n + 21561488819838000) 2 2 2 / 2 (2 n - 7) (n - 2) (n - 3) b(n - 4) / (n (4 n - 3) (3 n - 1) (3 n - 2) / 2 (4 n - 1) %1 (2 n - 1) ) 14 13 %1 := 639029665947318048 n - 18334623430327630608 n 12 11 + 242017376384513374104 n - 1948126830566463997228 n 10 9 + 10683676237670357758328 n - 42227126976668318125224 n 8 7 + 124051799176661063081400 n - 275171941204132659304527 n 6 5 + 463148432527246189718019 n - 588636459311606326921775 n 4 3 + 556107994882603121100931 n - 378711745571406327498834 n 2 + 175748657969442288305034 n - 49751730089227609240788 n + 6482268968853371901120 and in Maple notation b(n) = 1/108*(1773202522138592223040128*n^22-57968383229470970097940800*n^21+ 887350263343019533619913216*n^20-8456888142633929237962932064*n^19+ 56283876727733765752749935112*n^18-278109560393144559014410568836*n^17+ 1058932888815707720474433818900*n^16-3183100668992380937441875509256*n^15+ 7677255208641113541703728593744*n^14-15019140486763404798373859483003*n^13+ 23997613092764794651370342959556*n^12-31435161897450627453521248241825*n^11+ 33794565042858297182576753524924*n^10-29770539630405502463187380040071*n^9+ 21397098018899382044554800194164*n^8-12452457741046333450562147983371*n^7+ 5799305179305846117873035976052*n^6-2123726263111614035468525389710*n^5+ 595779869185627487035029176124*n^4-123031857893879428466892291864*n^3+ 17529038371951857657287020080*n^2-1529540763531984398189131200*n+ 61164659259102399960960000)/n^2/(4*n-3)/(3*n-1)/(3*n-2)/(4*n-1)/( 639029665947318048*n^14-18334623430327630608*n^13+242017376384513374104*n^12-\ 1948126830566463997228*n^11+10683676237670357758328*n^10-\ 42227126976668318125224*n^9+124051799176661063081400*n^8-\ 275171941204132659304527*n^7+463148432527246189718019*n^6-\ 588636459311606326921775*n^5+556107994882603121100931*n^4-\ 378711745571406327498834*n^3+175748657969442288305034*n^2-\ 49751730089227609240788*n+6482268968853371901120)/(2*n-1)^2*b(n-1)+1/18*(2*n-3) *(30259698266906326074819456*n^21-974100520011648253216591872*n^20+ 14642526759952397444831748480*n^19-136584303069243712851959905248*n^18+ 886155394517793048834614455656*n^17-4247989178691186268353778021768*n^16+ 15600598894277331233878611851104*n^15-44910623758824651837181891313800*n^14+ 102844961970576342040207311272068*n^13-189030088283080695815622606640938*n^12+ 280144601672964417743740822768799*n^11-335070301686498614803334562958530*n^10+ 322647379911762224561098019583115*n^9-248668638218303614924866255820688*n^8+ 151934618964420348980407511839097*n^7-72568166810799071926515478141482*n^6+ 26564860267833594568999432043005*n^5-7248651801037048460687141137634*n^4+ 1415968703080770455911648534420*n^3-185927795943276484548438053640*n^2+ 14652571406718771356195577600*n-525191635382693403358080000)/n^2/(4*n-3)/(3*n-1 )/(3*n-2)/(4*n-1)/(639029665947318048*n^14-18334623430327630608*n^13+ 242017376384513374104*n^12-1948126830566463997228*n^11+10683676237670357758328* n^10-42227126976668318125224*n^9+124051799176661063081400*n^8-\ 275171941204132659304527*n^7+463148432527246189718019*n^6-\ 588636459311606326921775*n^5+556107994882603121100931*n^4-\ 378711745571406327498834*n^3+175748657969442288305034*n^2-\ 49751730089227609240788*n+6482268968853371901120)/(2*n-1)^2*b(n-2)+1/9*(2*n-3)* (2*n-5)*(180101564931928329376128*n^18-4807153799245961441284032*n^17+ 58883933773094347527314880*n^16-439152860714772227893817984*n^15+ 2230927533690048272526574888*n^14-8183331164817549768520253252*n^13+ 22419130077139891605177862156*n^12-46804703028238380346930764828*n^11+ 75310626117753087284698375058*n^10-93848684664415700238198661727*n^9+ 90528074034717438503367828518*n^8-67221177324360101270248707070*n^7+ 38006234848095368197742784584*n^6-16081404118748913572910517611*n^5+ 4962915567782881865188321318*n^4-1074969937225421035752299660*n^3+ 153767358522183936619842294*n^2-13003982416281808791728460*n+ 494581794236591351598000)*(n-2)^2/n^2/(4*n-3)/(3*n-1)/(3*n-2)/(4*n-1)/( 639029665947318048*n^14-18334623430327630608*n^13+242017376384513374104*n^12-\ 1948126830566463997228*n^11+10683676237670357758328*n^10-\ 42227126976668318125224*n^9+124051799176661063081400*n^8-\ 275171941204132659304527*n^7+463148432527246189718019*n^6-\ 588636459311606326921775*n^5+556107994882603121100931*n^4-\ 378711745571406327498834*n^3+175748657969442288305034*n^2-\ 49751730089227609240788*n+6482268968853371901120)/(2*n-1)^2*b(n-3)+1331/27*(2*n -3)*(2*n-5)*(639029665947318048*n^14-9388208107065177936*n^13+ 61818971391460118568*n^12-241412143113034925932*n^11+623394337356699491844*n^10 -1123435837985043239228*n^9+1453107080687995223652*n^8-1366126413770221539495*n ^7+934817253445320523170*n^6-461619297950334419264*n^5+161357471792728350120*n^ 4-38614124995148396137*n^3+5978770560190065750*n^2-538466543363273976*n+ 21561488819838000)*(2*n-7)^2*(n-2)^2*(n-3)^2/n^2/(4*n-3)/(3*n-1)/(3*n-2)/(4*n-1 )/(639029665947318048*n^14-18334623430327630608*n^13+242017376384513374104*n^12 -1948126830566463997228*n^11+10683676237670357758328*n^10-\ 42227126976668318125224*n^9+124051799176661063081400*n^8-\ 275171941204132659304527*n^7+463148432527246189718019*n^6-\ 588636459311606326921775*n^5+556107994882603121100931*n^4-\ 378711745571406327498834*n^3+175748657969442288305034*n^2-\ 49751730089227609240788*n+6482268968853371901120)/(2*n-1)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 25, b(2) = 1657, b(3) = 137458 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 22 21 a(n) = 1/108 (1773202522138592223040128 n - 57968383229470970097940800 n 20 19 + 887350263343019533619913216 n - 8456888142633929237962932064 n 18 17 + 56283876727733765752749935112 n - 278109560393144559014410568836 n 16 + 1058932888815707720474433818900 n 15 - 3183100668992380937441875509256 n 14 + 7677255208641113541703728593744 n 13 - 15019140486763404798373859483003 n 12 + 23997613092764794651370342959556 n 11 - 31435161897450627453521248241825 n 10 + 33794565042858297182576753524924 n 9 - 29770539630405502463187380040071 n 8 + 21397098018899382044554800194164 n 7 6 - 12452457741046333450562147983371 n + 5799305179305846117873035976052 n 5 4 - 2123726263111614035468525389710 n + 595779869185627487035029176124 n 3 2 - 123031857893879428466892291864 n + 17529038371951857657287020080 n - 1529540763531984398189131200 n + 61164659259102399960960000) a(n - 1) / 2 2 / (n (4 n - 3) (3 n - 1) (3 n - 2) (4 n - 1) %1 (2 n - 1) ) + 1/18 / 21 20 (2 n - 3) (30259698266906326074819456 n - 974100520011648253216591872 n 19 18 + 14642526759952397444831748480 n - 136584303069243712851959905248 n 17 16 + 886155394517793048834614455656 n - 4247989178691186268353778021768 n 15 + 15600598894277331233878611851104 n 14 - 44910623758824651837181891313800 n 13 + 102844961970576342040207311272068 n 12 - 189030088283080695815622606640938 n 11 + 280144601672964417743740822768799 n 10 - 335070301686498614803334562958530 n 9 + 322647379911762224561098019583115 n 8 - 248668638218303614924866255820688 n 7 + 151934618964420348980407511839097 n 6 - 72568166810799071926515478141482 n 5 4 + 26564860267833594568999432043005 n - 7248651801037048460687141137634 n 3 2 + 1415968703080770455911648534420 n - 185927795943276484548438053640 n + 14652571406718771356195577600 n - 525191635382693403358080000) a(n - 2) / 2 2 / (n (4 n - 3) (3 n - 1) (3 n - 2) (4 n - 1) %1 (2 n - 1) ) + 1/9 / 18 (2 n - 3) (2 n - 5) (180101564931928329376128 n 17 16 - 4807153799245961441284032 n + 58883933773094347527314880 n 15 14 - 439152860714772227893817984 n + 2230927533690048272526574888 n 13 12 - 8183331164817549768520253252 n + 22419130077139891605177862156 n 11 10 - 46804703028238380346930764828 n + 75310626117753087284698375058 n 9 8 - 93848684664415700238198661727 n + 90528074034717438503367828518 n 7 6 - 67221177324360101270248707070 n + 38006234848095368197742784584 n 5 4 - 16081404118748913572910517611 n + 4962915567782881865188321318 n 3 2 - 1074969937225421035752299660 n + 153767358522183936619842294 n 2 - 13003982416281808791728460 n + 494581794236591351598000) (n - 2) / 2 2 a(n - 3) / (n (4 n - 3) (3 n - 1) (3 n - 2) (4 n - 1) %1 (2 n - 1) ) + / 1331 14 13 ---- (2 n - 3) (2 n - 5) (639029665947318048 n - 9388208107065177936 n 27 12 11 + 61818971391460118568 n - 241412143113034925932 n 10 9 + 623394337356699491844 n - 1123435837985043239228 n 8 7 + 1453107080687995223652 n - 1366126413770221539495 n 6 5 + 934817253445320523170 n - 461619297950334419264 n 4 3 + 161357471792728350120 n - 38614124995148396137 n 2 + 5978770560190065750 n - 538466543363273976 n + 21561488819838000) 2 2 2 / 2 (2 n - 7) (n - 2) (n - 3) a(n - 4) / (n (4 n - 3) (3 n - 1) (3 n - 2) / 2 (4 n - 1) %1 (2 n - 1) ) 14 13 %1 := 639029665947318048 n - 18334623430327630608 n 12 11 + 242017376384513374104 n - 1948126830566463997228 n 10 9 + 10683676237670357758328 n - 42227126976668318125224 n 8 7 + 124051799176661063081400 n - 275171941204132659304527 n 6 5 + 463148432527246189718019 n - 588636459311606326921775 n 4 3 + 556107994882603121100931 n - 378711745571406327498834 n 2 + 175748657969442288305034 n - 49751730089227609240788 n + 6482268968853371901120 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.01979481601203801245757198625085631900265748752833782795918368168716205933\ 727223363736244603267985746 This constant is identified as, 2/111 ln(3) The implied delta is, -0.9401697853158025989823513299282604463715072866221295\ 064358737361301365053729048176045488687310363187 Since this is negative, there is no Apery-style irrationality proof of, 2/111 ln(3), but still a very fast way to compute it to many digits ----------------------- This took, 2.602, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ ) binomial(n, k) binomial(2 n, k) binomial(2 n + k, k) / ----- k By Shalosh B. Ekhad Let ----- \ b(n) = ) binomial(n, k) binomial(2 n, k) binomial(2 n + k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 6 5 4 3 2 b(n) = (6528 n - 30192 n + 55040 n - 50406 n + 24465 n - 5985 n + 585) / 2 2 2 b(n - 1) / (n (48 n - 126 n + 83) (2 n - 1) ) / 2 2 2 (48 n - 30 n + 5) (n - 1) (2 n - 3) b(n - 2) - ----------------------------------------------- 2 2 2 n (48 n - 126 n + 83) (2 n - 1) and in Maple notation b(n) = (6528*n^6-30192*n^5+55040*n^4-50406*n^3+24465*n^2-5985*n+585)/n^2/(48*n^ 2-126*n+83)/(2*n-1)^2*b(n-1)-(48*n^2-30*n+5)*(n-1)^2*(2*n-3)^2/n^2/(48*n^2-126* n+83)/(2*n-1)^2*b(n-2) Of course, the initial conditions are b(0) = 1, b(1) = 7 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 6 5 4 3 2 a(n) = (6528 n - 30192 n + 55040 n - 50406 n + 24465 n - 5985 n + 585) / 2 2 2 a(n - 1) / (n (48 n - 126 n + 83) (2 n - 1) ) / 2 2 2 (48 n - 30 n + 5) (n - 1) (2 n - 3) a(n - 2) - ----------------------------------------------- 2 2 2 n (48 n - 126 n + 83) (2 n - 1) but with the following simpler initial conditions a(0) = 0, a(1) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.14303774494332403795412305796921958167121303488754769023787462864087021481\ 76696412029242522278051048 2 Pi This constant is identified as, --- 69 The implied delta is, -0.0604626290026458773344125018774411738927566375995041\ 179493927032480636635250768256088955552559271054 2 Pi Since this is negative, there is no Apery-style irrationality proof of, ---, 69 but still a very fast way to compute it to many digits ----------------------- This took, 2.583, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(2 n, k) binomial(2 n + k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(2 n, k) binomial(2 n + k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 9 8 7 6 5 b(n) = (1251264 n - 13674528 n + 63396928 n - 162926864 n + 254659060 n 4 3 2 - 250004822 n + 153635704 n - 56901029 n + 11542632 n - 985950) / 2 2 9 8 7 b(n - 1) / (n %1 (2 n - 1) ) + (6344128 n - 75676384 n + 386498560 n / 6 5 4 3 - 1103148176 n + 1925222884 n - 2110111922 n + 1433966688 n 2 / 2 2 - 573324535 n + 120654852 n - 10381950) b(n - 2) / (n %1 (2 n - 1) ) / 5 4 3 2 2 - (5488 n - 21560 n + 31052 n - 19954 n + 5604 n - 585) (n - 2) 2 / 2 2 (2 n - 5) b(n - 3) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 5488 n - 49000 n + 172172 n - 297350 n + 252348 n - 84243 and in Maple notation b(n) = (1251264*n^9-13674528*n^8+63396928*n^7-162926864*n^6+254659060*n^5-\ 250004822*n^4+153635704*n^3-56901029*n^2+11542632*n-985950)/n^2/(5488*n^5-49000 *n^4+172172*n^3-297350*n^2+252348*n-84243)/(2*n-1)^2*b(n-1)+(6344128*n^9-\ 75676384*n^8+386498560*n^7-1103148176*n^6+1925222884*n^5-2110111922*n^4+ 1433966688*n^3-573324535*n^2+120654852*n-10381950)/n^2/(5488*n^5-49000*n^4+ 172172*n^3-297350*n^2+252348*n-84243)/(2*n-1)^2*b(n-2)-(5488*n^5-21560*n^4+ 31052*n^3-19954*n^2+5604*n-585)*(n-2)^2*(2*n-5)^2/n^2/(5488*n^5-49000*n^4+ 172172*n^3-297350*n^2+252348*n-84243)/(2*n-1)^2*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 13, b(2) = 441 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 9 8 7 6 5 a(n) = (1251264 n - 13674528 n + 63396928 n - 162926864 n + 254659060 n 4 3 2 - 250004822 n + 153635704 n - 56901029 n + 11542632 n - 985950) / 2 2 9 8 7 a(n - 1) / (n %1 (2 n - 1) ) + (6344128 n - 75676384 n + 386498560 n / 6 5 4 3 - 1103148176 n + 1925222884 n - 2110111922 n + 1433966688 n 2 / 2 2 - 573324535 n + 120654852 n - 10381950) a(n - 2) / (n %1 (2 n - 1) ) / 5 4 3 2 2 - (5488 n - 21560 n + 31052 n - 19954 n + 5604 n - 585) (n - 2) 2 / 2 2 (2 n - 5) a(n - 3) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 5488 n - 49000 n + 172172 n - 297350 n + 252348 n - 84243 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.06931471805599453094172321214581765680755001343602552541206800094933936219\ 696947156058633269964186875 This constant is identified as, 1/10 ln(2) The implied delta is, -0.5774941402332961270519299428725279698884689265527361\ 110795077025862981700034163965916252497829383424 Since this is negative, there is no Apery-style irrationality proof of, 1/10 ln(2), but still a very fast way to compute it to many digits ----------------------- This took, 1.103, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(2 n, k) binomial(2 n + k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(2 n, k) binomial(2 n + k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 9 8 7 6 b(n) = (11556864 n - 125528832 n + 578172192 n - 1475901768 n 5 4 3 2 + 2291922084 n - 2237526190 n + 1370127087 n - 507379955 n / 2 2 9 + 103388163 n - 8896050) b(n - 1) / (n %1 (2 n - 1) ) + (243175680 n / 8 7 6 5 - 2884511520 n + 14627584464 n - 41370656304 n + 71342328120 n 4 3 2 - 76949350618 n + 51145289521 n - 19812569744 n + 3985645235 n / 2 2 - 324939750) b(n - 2) / (n %1 (2 n - 1) ) - 2 / 5 4 3 2 2 (40128 n - 154968 n + 217692 n - 134698 n + 35731 n - 3495) (n - 2) 2 / 2 2 (2 n - 5) b(n - 3) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 40128 n - 355608 n + 1238844 n - 2118862 n + 1778715 n - 586712 and in Maple notation b(n) = (11556864*n^9-125528832*n^8+578172192*n^7-1475901768*n^6+2291922084*n^5-\ 2237526190*n^4+1370127087*n^3-507379955*n^2+103388163*n-8896050)/n^2/(40128*n^5 -355608*n^4+1238844*n^3-2118862*n^2+1778715*n-586712)/(2*n-1)^2*b(n-1)+( 243175680*n^9-2884511520*n^8+14627584464*n^7-41370656304*n^6+71342328120*n^5-\ 76949350618*n^4+51145289521*n^3-19812569744*n^2+3985645235*n-324939750)/n^2/( 40128*n^5-355608*n^4+1238844*n^3-2118862*n^2+1778715*n-586712)/(2*n-1)^2*b(n-2) -2*(40128*n^5-154968*n^4+217692*n^3-134698*n^2+35731*n-3495)*(n-2)^2*(2*n-5)^2/ n^2/(40128*n^5-355608*n^4+1238844*n^3-2118862*n^2+1778715*n-586712)/(2*n-1)^2*b (n-3) Of course, the initial conditions are b(0) = 1, b(1) = 19, b(2) = 931 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 9 8 7 6 a(n) = (11556864 n - 125528832 n + 578172192 n - 1475901768 n 5 4 3 2 + 2291922084 n - 2237526190 n + 1370127087 n - 507379955 n / 2 2 9 + 103388163 n - 8896050) a(n - 1) / (n %1 (2 n - 1) ) + (243175680 n / 8 7 6 5 - 2884511520 n + 14627584464 n - 41370656304 n + 71342328120 n 4 3 2 - 76949350618 n + 51145289521 n - 19812569744 n + 3985645235 n / 2 2 - 324939750) a(n - 2) / (n %1 (2 n - 1) ) - 2 / 5 4 3 2 2 (40128 n - 154968 n + 217692 n - 134698 n + 35731 n - 3495) (n - 2) 2 / 2 2 (2 n - 5) a(n - 3) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 40128 n - 355608 n + 1238844 n - 2118862 n + 1778715 n - 586712 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.04225431879492729582289404757394329633259579068549036352825747437067285743\ 148496026436983672360508034 This constant is identified as, 1/26 ln(3) The implied delta is, -0.7437866765777699020226367843238357979070319837434721\ 161166140012974924355327685688630091692370339409 Since this is negative, there is no Apery-style irrationality proof of, 1/26 ln(3), but still a very fast way to compute it to many digits ----------------------- This took, 0.952, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 ) binomial(n, k) binomial(2 n + k, k) / ----- k By Shalosh B. Ekhad Let ----- \ 2 b(n) = ) binomial(n, k) binomial(2 n + k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 8 7 6 5 b(n) = 1/32 (41891364 n - 371204031 n + 1361660056 n - 2687630688 n 4 3 2 + 3107108770 n - 2145515097 n + 861999250 n - 184068984 n + 16035840) / 2 2 8 7 b(n - 1) / (n %1 (2 n - 1) ) + 1/16 (1798416 n - 17734380 n / 6 5 4 3 2 + 73145710 n - 163759365 n + 215908696 n - 169951719 n + 77112394 n / 2 2 - 18266400 n + 1728000) b(n - 2) / (n %1 (2 n - 1) ) + 1/8 (n - 1) / 4 3 2 2 (n - 2) (6516 n - 18643 n + 17957 n - 6768 n + 864) (2 n - 5) b(n - 3) / 2 2 / (n %1 (2 n - 1) ) / 4 3 2 %1 := 6516 n - 44707 n + 112982 n - 124675 n + 50748 and in Maple notation b(n) = 1/32*(41891364*n^8-371204031*n^7+1361660056*n^6-2687630688*n^5+ 3107108770*n^4-2145515097*n^3+861999250*n^2-184068984*n+16035840)/n^2/(6516*n^4 -44707*n^3+112982*n^2-124675*n+50748)/(2*n-1)^2*b(n-1)+1/16*(1798416*n^8-\ 17734380*n^7+73145710*n^6-163759365*n^5+215908696*n^4-169951719*n^3+77112394*n^ 2-18266400*n+1728000)/n^2/(6516*n^4-44707*n^3+112982*n^2-124675*n+50748)/(2*n-1 )^2*b(n-2)+1/8*(n-1)*(n-2)*(6516*n^4-18643*n^3+17957*n^2-6768*n+864)*(2*n-5)^2/ n^2/(6516*n^4-44707*n^3+112982*n^2-124675*n+50748)/(2*n-1)^2*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 10, b(2) = 276 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 8 7 6 5 a(n) = 1/32 (41891364 n - 371204031 n + 1361660056 n - 2687630688 n 4 3 2 + 3107108770 n - 2145515097 n + 861999250 n - 184068984 n + 16035840) / 2 2 8 7 a(n - 1) / (n %1 (2 n - 1) ) + 1/16 (1798416 n - 17734380 n / 6 5 4 3 2 + 73145710 n - 163759365 n + 215908696 n - 169951719 n + 77112394 n / 2 2 - 18266400 n + 1728000) a(n - 2) / (n %1 (2 n - 1) ) + 1/8 (n - 1) / 4 3 2 2 (n - 2) (6516 n - 18643 n + 17957 n - 6768 n + 864) (2 n - 5) a(n - 3) / 2 2 / (n %1 (2 n - 1) ) / 4 3 2 %1 := 6516 n - 44707 n + 112982 n - 124675 n + 50748 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.09991310710773986442050192741739482062349551486273949608946738875580448604\ 968572477201633542290719820 16 This constant is identified as, --- ln(2) 111 The implied delta is, -0.1321678012603518355735568199912384974727950070737104\ 121460239903099652396915570435246949844858851619 Since this is negative, there is no Apery-style irrationality proof of, 16 --- ln(2), but still a very fast way to compute it to many digits 111 ----------------------- This took, 1.622, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(2 n + k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(2 n + k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 9 8 7 6 b(n) = 1/9 (563604480 n - 6105715200 n + 28125939960 n - 71986379580 n 5 4 3 2 + 112275415680 n - 110112791189 n + 67585333713 n - 24940629066 n / 2 2 + 5009653328 n - 417673260) b(n - 1) / (n %1 (2 n - 1) ) + 1/9 ( / 9 8 7 6 5 9784800 n - 115786800 n + 587398680 n - 1667374800 n + 2897902980 n 4 3 2 - 3169852065 n + 2159187255 n - 873360970 n + 188954668 n - 16674660) / 2 2 b(n - 2) / (n %1 (2 n - 1) ) + 1/27 / 5 4 3 2 2 (163080 n - 625140 n + 890340 n - 580535 n + 172665 n - 18954) (n - 2) 2 / 2 2 (2 n - 5) b(n - 3) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 163080 n - 1440540 n + 5021700 n - 8633195 n + 7320715 n - 2450714 and in Maple notation b(n) = 1/9*(563604480*n^9-6105715200*n^8+28125939960*n^7-71986379580*n^6+ 112275415680*n^5-110112791189*n^4+67585333713*n^3-24940629066*n^2+5009653328*n-\ 417673260)/n^2/(163080*n^5-1440540*n^4+5021700*n^3-8633195*n^2+7320715*n-\ 2450714)/(2*n-1)^2*b(n-1)+1/9*(9784800*n^9-115786800*n^8+587398680*n^7-\ 1667374800*n^6+2897902980*n^5-3169852065*n^4+2159187255*n^3-873360970*n^2+ 188954668*n-16674660)/n^2/(163080*n^5-1440540*n^4+5021700*n^3-8633195*n^2+ 7320715*n-2450714)/(2*n-1)^2*b(n-2)+1/27*(163080*n^5-625140*n^4+890340*n^3-\ 580535*n^2+172665*n-18954)*(n-2)^2*(2*n-5)^2/n^2/(163080*n^5-1440540*n^4+ 5021700*n^3-8633195*n^2+7320715*n-2450714)/(2*n-1)^2*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 19, b(2) = 1001 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 9 8 7 6 a(n) = 1/9 (563604480 n - 6105715200 n + 28125939960 n - 71986379580 n 5 4 3 2 + 112275415680 n - 110112791189 n + 67585333713 n - 24940629066 n / 2 2 + 5009653328 n - 417673260) a(n - 1) / (n %1 (2 n - 1) ) + 1/9 ( / 9 8 7 6 5 9784800 n - 115786800 n + 587398680 n - 1667374800 n + 2897902980 n 4 3 2 - 3169852065 n + 2159187255 n - 873360970 n + 188954668 n - 16674660) / 2 2 a(n - 2) / (n %1 (2 n - 1) ) + 1/27 / 5 4 3 2 2 (163080 n - 625140 n + 890340 n - 580535 n + 172665 n - 18954) (n - 2) 2 / 2 2 (2 n - 5) a(n - 3) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 163080 n - 1440540 n + 5021700 n - 8633195 n + 7320715 n - 2450714 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.05263248999480979958368439479585301292040260304561222757489609015332076021\ 019559992417940168782433574 27 27 This constant is identified as, --- ln(3) - --- ln(2) 208 208 The implied delta is, -0.1364740049899604076897001960071826381031599171335240\ 797860795340889788013711902819034083815190522700 Since this is negative, there is no Apery-style irrationality proof of, 27 27 --- ln(3) - --- ln(2), 208 208 but still a very fast way to compute it to many digits ----------------------- This took, 2.665, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(2 n + k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(2 n + k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 9 8 7 b(n) = 3/256 (128025219924 n - 1385415772749 n + 6373308506610 n 6 5 4 - 16287756246798 n + 25365448463856 n - 24842015353949 n 3 2 + 15229155844842 n - 5614512736616 n + 1126975500544 n - 93923573760) / 2 2 9 8 b(n - 1) / (n %1 (2 n - 1) ) + 3/128 (201268368 n - 2379279636 n / 7 6 5 4 + 12072305802 n - 34273632323 n + 59476361594 n - 64703822221 n 3 2 + 43549843900 n - 17261463508 n + 3637757632 n - 312238080) b(n - 2) / 2 2 / (n %1 (2 n - 1) ) + 1/64 / 5 4 3 2 (2648268 n - 10120167 n + 14323872 n - 9252727 n + 2726838 n - 297216) 2 2 / 2 2 (n - 2) (2 n - 5) b(n - 3) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 2648268 n - 23361507 n + 81287220 n - 139428025 n + 117925916 n - 39369088 and in Maple notation b(n) = 3/256*(128025219924*n^9-1385415772749*n^8+6373308506610*n^7-\ 16287756246798*n^6+25365448463856*n^5-24842015353949*n^4+15229155844842*n^3-\ 5614512736616*n^2+1126975500544*n-93923573760)/n^2/(2648268*n^5-23361507*n^4+ 81287220*n^3-139428025*n^2+117925916*n-39369088)/(2*n-1)^2*b(n-1)+3/128*( 201268368*n^9-2379279636*n^8+12072305802*n^7-34273632323*n^6+59476361594*n^5-\ 64703822221*n^4+43549843900*n^3-17261463508*n^2+3637757632*n-312238080)/n^2/( 2648268*n^5-23361507*n^4+81287220*n^3-139428025*n^2+117925916*n-39369088)/(2*n-\ 1)^2*b(n-2)+1/64*(2648268*n^5-10120167*n^4+14323872*n^3-9252727*n^2+2726838*n-\ 297216)*(n-2)^2*(2*n-5)^2/n^2/(2648268*n^5-23361507*n^4+81287220*n^3-139428025* n^2+117925916*n-39369088)/(2*n-1)^2*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 28, b(2) = 2176 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 9 8 7 a(n) = 3/256 (128025219924 n - 1385415772749 n + 6373308506610 n 6 5 4 - 16287756246798 n + 25365448463856 n - 24842015353949 n 3 2 + 15229155844842 n - 5614512736616 n + 1126975500544 n - 93923573760) / 2 2 9 8 a(n - 1) / (n %1 (2 n - 1) ) + 3/128 (201268368 n - 2379279636 n / 7 6 5 4 + 12072305802 n - 34273632323 n + 59476361594 n - 64703822221 n 3 2 + 43549843900 n - 17261463508 n + 3637757632 n - 312238080) a(n - 2) / 2 2 / (n %1 (2 n - 1) ) + 1/64 / 5 4 3 2 (2648268 n - 10120167 n + 14323872 n - 9252727 n + 2726838 n - 297216) 2 2 / 2 2 (n - 2) (2 n - 5) a(n - 3) / (n %1 (2 n - 1) ) / 5 4 3 2 %1 := 2648268 n - 23361507 n + 81287220 n - 139428025 n + 117925916 n - 39369088 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.03571610598819394637460720927954404581226890688158430187473637994637196672\ 382143495177322511631956998 256 128 This constant is identified as, ---- ln(2) - ---- ln(3) 1031 1031 The implied delta is, -0.1423199514723936132909969177794104190180969567811539\ 994653659333517531122444784456608821515589480665 Since this is negative, there is no Apery-style irrationality proof of, 256 128 ---- ln(2) - ---- ln(3), 1031 1031 but still a very fast way to compute it to many digits ----------------------- This took, 2.312, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ ) binomial(n, k) binomial(2 n + k, k) binomial(2 n + 2 k, k) / ----- k By Shalosh B. Ekhad Let ----- \ b(n) = ) binomial(n, k) binomial(2 n + k, k) binomial(2 n + 2 k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 7 6 5 4 3 b(n) = 1/5 (105090 n - 928521 n + 3299409 n - 6053082 n + 6126321 n 2 / 2 - 3376667 n + 924275 n - 94875) b(n - 1) / ((2 n - 1) %1 n ) + 1/5 ( / 8 7 6 5 4 3 17980 n - 203812 n + 977389 n - 2573355 n + 4031675 n - 3797150 n 2 / 2 + 2056516 n - 567425 n + 58875) b(n - 2) / (n (2 n - 1) (2 n - 3) %1) / 4 3 2 2 2 (155 n - 517 n + 574 n - 235 n + 30) (n - 2) (2 n - 5) b(n - 3) + 1/5 -------------------------------------------------------------------- 2 n (2 n - 1) (2 n - 3) %1 4 3 2 %1 := 155 n - 1137 n + 3055 n - 3554 n + 1511 and in Maple notation b(n) = 1/5*(105090*n^7-928521*n^6+3299409*n^5-6053082*n^4+6126321*n^3-3376667*n ^2+924275*n-94875)/(2*n-1)/(155*n^4-1137*n^3+3055*n^2-3554*n+1511)/n^2*b(n-1)+1 /5*(17980*n^8-203812*n^7+977389*n^6-2573355*n^5+4031675*n^4-3797150*n^3+2056516 *n^2-567425*n+58875)/n^2/(2*n-1)/(2*n-3)/(155*n^4-1137*n^3+3055*n^2-3554*n+1511 )*b(n-2)+1/5*(155*n^4-517*n^3+574*n^2-235*n+30)*(n-2)^2*(2*n-5)^2/n^2/(2*n-1)/( 2*n-3)/(155*n^4-1137*n^3+3055*n^2-3554*n+1511)*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 13, b(2) = 481 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 7 6 5 4 3 a(n) = 1/5 (105090 n - 928521 n + 3299409 n - 6053082 n + 6126321 n 2 / 2 - 3376667 n + 924275 n - 94875) a(n - 1) / ((2 n - 1) %1 n ) + 1/5 ( / 8 7 6 5 4 3 17980 n - 203812 n + 977389 n - 2573355 n + 4031675 n - 3797150 n 2 / 2 + 2056516 n - 567425 n + 58875) a(n - 2) / (n (2 n - 1) (2 n - 3) %1) / 4 3 2 2 2 (155 n - 517 n + 574 n - 235 n + 30) (n - 2) (2 n - 5) a(n - 3) + 1/5 -------------------------------------------------------------------- 2 n (2 n - 1) (2 n - 3) %1 4 3 2 %1 := 155 n - 1137 n + 3055 n - 3554 n + 1511 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.07685873945785786408739879159832239739724476473933743834946537571295735990\ 377356452883754726736070632 The implied delta is, -0.0855134697182789664679060265268265421399827307430878\ 568051287775347370551492004742090097977649003283 Since this is negative, there is no Apery-style irrationality proof of, 0.076\ 858739457857864087398791598322397397244764739337438349465375712957359903\ 77356452883754726736070632, but still a very fast way to compute it to many digits ----------------------- This took, 6.392, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(2 n + k, k) binomial(2 n + 2 k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(2 n + k, k) binomial(2 n + 2 k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 7 6 5 4 3 b(n) = 1/9 (671842 n - 5929065 n + 21044261 n - 38570276 n + 39014381 n 2 / 2 - 21504891 n + 5892183 n - 606285) b(n - 1) / ((2 n - 1) %1 n ) + 1/9 ( / 8 7 6 5 4 3 69052 n - 782020 n + 3744681 n - 9838047 n + 15366957 n - 14414302 n 2 / 2 + 7765198 n - 2129013 n + 220185) b(n - 2) / (n (2 n - 1) (2 n - 3) %1 / ) + 4 3 2 2 2 (283 n - 941 n + 1040 n - 423 n + 54) (n - 2) (2 n - 5) b(n - 3) 1/9 --------------------------------------------------------------------- 2 n (2 n - 1) (2 n - 3) %1 4 3 2 %1 := 283 n - 2073 n + 5561 n - 6458 n + 2741 and in Maple notation b(n) = 1/9*(671842*n^7-5929065*n^6+21044261*n^5-38570276*n^4+39014381*n^3-\ 21504891*n^2+5892183*n-606285)/(2*n-1)/(283*n^4-2073*n^3+5561*n^2-6458*n+2741)/ n^2*b(n-1)+1/9*(69052*n^8-782020*n^7+3744681*n^6-9838047*n^5+15366957*n^4-\ 14414302*n^3+7765198*n^2-2129013*n+220185)/n^2/(2*n-1)/(2*n-3)/(283*n^4-2073*n^ 3+5561*n^2-6458*n+2741)*b(n-2)+1/9*(283*n^4-941*n^3+1040*n^2-423*n+54)*(n-2)^2* (2*n-5)^2/n^2/(2*n-1)/(2*n-3)/(283*n^4-2073*n^3+5561*n^2-6458*n+2741)*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 25, b(2) = 1801 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 7 6 5 4 3 a(n) = 1/9 (671842 n - 5929065 n + 21044261 n - 38570276 n + 39014381 n 2 / 2 - 21504891 n + 5892183 n - 606285) a(n - 1) / ((2 n - 1) %1 n ) + 1/9 ( / 8 7 6 5 4 3 69052 n - 782020 n + 3744681 n - 9838047 n + 15366957 n - 14414302 n 2 / 2 + 7765198 n - 2129013 n + 220185) a(n - 2) / (n (2 n - 1) (2 n - 3) %1 / ) + 4 3 2 2 2 (283 n - 941 n + 1040 n - 423 n + 54) (n - 2) (2 n - 5) a(n - 3) 1/9 --------------------------------------------------------------------- 2 n (2 n - 1) (2 n - 3) %1 4 3 2 %1 := 283 n - 2073 n + 5561 n - 6458 n + 2741 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.03998926041691992169714800700720249431204808467463011081465461593231117049\ 825161820803057655748569351 This constant is identified as, 3/52 ln(2) The implied delta is, -0.0698752982845850019438738883822275673980536310869105\ 366290884748929148133214400753376868128909392394 Since this is negative, there is no Apery-style irrationality proof of, 3/52 ln(2), but still a very fast way to compute it to many digits ----------------------- This took, 2.290, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(2 n + k, k) binomial(2 n + 2 k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(2 n + k, k) binomial(2 n + 2 k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 7 6 5 4 b(n) = 1/13 (2093634 n - 18468297 n + 65521593 n - 120043470 n 3 2 / + 121395945 n - 66912347 n + 18339035 n - 1888575) b(n - 1) / ( / 2 8 7 6 5 (2 n - 1) %1 n ) + 1/13 (152892 n - 1730916 n + 8284005 n - 21747027 n 4 3 2 + 33932559 n - 31783294 n + 17089928 n - 4675177 n + 483015) b(n - 2) / 2 / (n (2 n - 1) (2 n - 3) %1) + / 4 3 2 2 2 (411 n - 1365 n + 1506 n - 611 n + 78) (n - 2) (2 n - 5) b(n - 3) 1/13 ---------------------------------------------------------------------- 2 n (2 n - 1) (2 n - 3) %1 4 3 2 %1 := 411 n - 3009 n + 8067 n - 9362 n + 3971 and in Maple notation b(n) = 1/13*(2093634*n^7-18468297*n^6+65521593*n^5-120043470*n^4+121395945*n^3-\ 66912347*n^2+18339035*n-1888575)/(2*n-1)/(411*n^4-3009*n^3+8067*n^2-9362*n+3971 )/n^2*b(n-1)+1/13*(152892*n^8-1730916*n^7+8284005*n^6-21747027*n^5+33932559*n^4 -31783294*n^3+17089928*n^2-4675177*n+483015)/n^2/(2*n-1)/(2*n-3)/(411*n^4-3009* n^3+8067*n^2-9362*n+3971)*b(n-2)+1/13*(411*n^4-1365*n^3+1506*n^2-611*n+78)*(n-2 )^2*(2*n-5)^2/n^2/(2*n-1)/(2*n-3)/(411*n^4-3009*n^3+8067*n^2-9362*n+3971)*b(n-3 ) Of course, the initial conditions are b(0) = 1, b(1) = 37, b(2) = 3961 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 7 6 5 4 a(n) = 1/13 (2093634 n - 18468297 n + 65521593 n - 120043470 n 3 2 / + 121395945 n - 66912347 n + 18339035 n - 1888575) a(n - 1) / ( / 2 8 7 6 5 (2 n - 1) %1 n ) + 1/13 (152892 n - 1730916 n + 8284005 n - 21747027 n 4 3 2 + 33932559 n - 31783294 n + 17089928 n - 4675177 n + 483015) a(n - 2) / 2 / (n (2 n - 1) (2 n - 3) %1) + / 4 3 2 2 2 (411 n - 1365 n + 1506 n - 611 n + 78) (n - 2) (2 n - 5) a(n - 3) 1/13 ---------------------------------------------------------------------- 2 n (2 n - 1) (2 n - 3) %1 4 3 2 %1 := 411 n - 3009 n + 8067 n - 9362 n + 3971 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.02702351667256146460142264421639296119425563606429443859521418350951418291\ 647948790030521301167253317 The implied delta is, -0.0628187094416363735434321807538920273377947445279044\ 102488017395998266473362360991745629260809079320 Since this is negative, there is no Apery-style irrationality proof of, 0.027\ 023516672561464601422644216392961194255636064294438595214183509514182916\ 47948790030521301167253317, but still a very fast way to compute it to many digits ----------------------- This took, 6.697, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ ) binomial(n, k) binomial(2 n, k) binomial(2 n + 2 k, k) / ----- k By Shalosh B. Ekhad Let ----- \ b(n) = ) binomial(n, k) binomial(2 n, k) binomial(2 n + 2 k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 22 21 b(n) = 1/12 (1639513016779632768 n - 53910416451510119232 n 20 19 + 830072450788236868992 n - 7956851638960649880160 n 18 17 + 53252041472084137404456 n - 264499500376448891889892 n 16 15 + 1011741506944383210720764 n - 3052469750624653163486632 n 14 13 + 7379931977041839436468616 n - 14447065971290520540737567 n 12 11 + 23045427368838340517524412 n - 30047624432061028761258221 n 10 9 + 32030823942973132138119268 n - 27848322173621377316531387 n 8 7 + 19643658815401034030830276 n - 11147248503722087490065967 n 6 5 + 5026193678411479875215572 n - 1768927863123393877442022 n 4 3 + 473603880303164077794156 n - 92809014800222694163320 n 2 + 12506052865184969364720 n - 1032238090527948561600 n / 2 + 39233793865875840000) b(n - 1) / (n (4 n - 3) (3 n - 1) (3 n - 2) / 2 21 (4 n - 1) %1 (2 n - 1) ) + 1/2 (2 n - 3) (670693534593699456 n 20 19 - 21718377351528834816 n + 328596081210820234368 n 18 17 - 3087193851575859029664 n + 20189524303044247378296 n 16 15 - 97642750906935044479560 n + 362149002268971676100480 n 14 13 - 1054165113426369325409456 n + 2444399523838901180103332 n 12 11 - 4557001780129530541567722 n + 6863724776836985779769611 n 10 9 - 8363519788943343941603410 n + 8228568660966128691821879 n 8 7 - 6502896685926921575181112 n + 4091957495643413813726221 n 6 5 - 2023708901766177331071906 n + 772168708486260366718577 n 4 3 - 221389000949527146674090 n + 45872596076056070518820 n 2 - 6455684199018518100264 n + 550519917163258728960 n / 2 - 21456613457693568000) b(n - 2) / (n (4 n - 3) (3 n - 1) (3 n - 2) / 2 18 (4 n - 1) %1 (2 n - 1) ) + (2 n - 3) (2 n - 5) (10997332632390528 n 17 16 - 295629956641658304 n + 3650771351387965248 n 15 14 - 27481211957287202112 n + 141099108420654830184 n 13 12 - 523928388876001820308 n + 1455685020034093887228 n 11 10 - 3088845481019532568372 n + 5064726784830846944946 n 9 8 - 6451907413647681669307 n + 6386544591741674491950 n 7 6 - 4889355966791524986830 n + 2866706330281325648064 n 5 4 - 1266924355365713970311 n + 411978257627804606334 n 3 2 - 95008211263362452396 n + 14635676564635537038 n 2 / 2 - 1346881942368923580 n + 56002064767110000) (n - 2) b(n - 3) / (n / 2 (4 n - 3) (3 n - 1) (3 n - 2) (4 n - 1) %1 (2 n - 1) ) + 9 (2 n - 3) 14 13 12 (2 n - 5) (5989832588448 n - 89140503841488 n + 595774252740648 n 11 10 9 - 2367022104074924 n + 6235750221135948 n - 11502479919483460 n 8 7 6 + 15289662094565532 n - 14845167670943811 n + 10555194555981042 n 5 4 3 - 5457214243694632 n + 2016044165229840 n - 515640016794845 n 2 2 2 + 86386787952222 n - 8511914500920 n + 374445126000) (2 n - 7) (n - 2) 2 / 2 (n - 3) b(n - 4) / (n (4 n - 3) (3 n - 1) (3 n - 2) (4 n - 1) %1 / 2 (2 n - 1) ) 14 13 12 %1 := 5989832588448 n - 172998160079760 n + 2299675568228760 n 11 10 9 - 18649571498793836 n + 103084100566544896 n - 410843638546643136 n 8 7 6 + 1217598939307121952 n - 2726034480616051467 n + 4633309960045658727 n 5 4 3 - 5949612399085521619 n + 5682064097238550343 n - 3913859601722074602 n 2 + 1838192079866743314 n - 526952131079799780 n + 69570352728653760 and in Maple notation b(n) = 1/12*(1639513016779632768*n^22-53910416451510119232*n^21+ 830072450788236868992*n^20-7956851638960649880160*n^19+53252041472084137404456* n^18-264499500376448891889892*n^17+1011741506944383210720764*n^16-\ 3052469750624653163486632*n^15+7379931977041839436468616*n^14-\ 14447065971290520540737567*n^13+23045427368838340517524412*n^12-\ 30047624432061028761258221*n^11+32030823942973132138119268*n^10-\ 27848322173621377316531387*n^9+19643658815401034030830276*n^8-\ 11147248503722087490065967*n^7+5026193678411479875215572*n^6-\ 1768927863123393877442022*n^5+473603880303164077794156*n^4-\ 92809014800222694163320*n^3+12506052865184969364720*n^2-1032238090527948561600* n+39233793865875840000)/n^2/(4*n-3)/(3*n-1)/(3*n-2)/(4*n-1)/(5989832588448*n^14 -172998160079760*n^13+2299675568228760*n^12-18649571498793836*n^11+ 103084100566544896*n^10-410843638546643136*n^9+1217598939307121952*n^8-\ 2726034480616051467*n^7+4633309960045658727*n^6-5949612399085521619*n^5+ 5682064097238550343*n^4-3913859601722074602*n^3+1838192079866743314*n^2-\ 526952131079799780*n+69570352728653760)/(2*n-1)^2*b(n-1)+1/2*(2*n-3)*( 670693534593699456*n^21-21718377351528834816*n^20+328596081210820234368*n^19-\ 3087193851575859029664*n^18+20189524303044247378296*n^17-\ 97642750906935044479560*n^16+362149002268971676100480*n^15-\ 1054165113426369325409456*n^14+2444399523838901180103332*n^13-\ 4557001780129530541567722*n^12+6863724776836985779769611*n^11-\ 8363519788943343941603410*n^10+8228568660966128691821879*n^9-\ 6502896685926921575181112*n^8+4091957495643413813726221*n^7-\ 2023708901766177331071906*n^6+772168708486260366718577*n^5-\ 221389000949527146674090*n^4+45872596076056070518820*n^3-6455684199018518100264 *n^2+550519917163258728960*n-21456613457693568000)/n^2/(4*n-3)/(3*n-1)/(3*n-2)/ (4*n-1)/(5989832588448*n^14-172998160079760*n^13+2299675568228760*n^12-\ 18649571498793836*n^11+103084100566544896*n^10-410843638546643136*n^9+ 1217598939307121952*n^8-2726034480616051467*n^7+4633309960045658727*n^6-\ 5949612399085521619*n^5+5682064097238550343*n^4-3913859601722074602*n^3+ 1838192079866743314*n^2-526952131079799780*n+69570352728653760)/(2*n-1)^2*b(n-2 )+(2*n-3)*(2*n-5)*(10997332632390528*n^18-295629956641658304*n^17+ 3650771351387965248*n^16-27481211957287202112*n^15+141099108420654830184*n^14-\ 523928388876001820308*n^13+1455685020034093887228*n^12-3088845481019532568372*n ^11+5064726784830846944946*n^10-6451907413647681669307*n^9+ 6386544591741674491950*n^8-4889355966791524986830*n^7+2866706330281325648064*n^ 6-1266924355365713970311*n^5+411978257627804606334*n^4-95008211263362452396*n^3 +14635676564635537038*n^2-1346881942368923580*n+56002064767110000)*(n-2)^2/n^2/ (4*n-3)/(3*n-1)/(3*n-2)/(4*n-1)/(5989832588448*n^14-172998160079760*n^13+ 2299675568228760*n^12-18649571498793836*n^11+103084100566544896*n^10-\ 410843638546643136*n^9+1217598939307121952*n^8-2726034480616051467*n^7+ 4633309960045658727*n^6-5949612399085521619*n^5+5682064097238550343*n^4-\ 3913859601722074602*n^3+1838192079866743314*n^2-526952131079799780*n+ 69570352728653760)/(2*n-1)^2*b(n-3)+9*(2*n-3)*(2*n-5)*(5989832588448*n^14-\ 89140503841488*n^13+595774252740648*n^12-2367022104074924*n^11+6235750221135948 *n^10-11502479919483460*n^9+15289662094565532*n^8-14845167670943811*n^7+ 10555194555981042*n^6-5457214243694632*n^5+2016044165229840*n^4-515640016794845 *n^3+86386787952222*n^2-8511914500920*n+374445126000)*(2*n-7)^2*(n-2)^2*(n-3)^2 /n^2/(4*n-3)/(3*n-1)/(3*n-2)/(4*n-1)/(5989832588448*n^14-172998160079760*n^13+ 2299675568228760*n^12-18649571498793836*n^11+103084100566544896*n^10-\ 410843638546643136*n^9+1217598939307121952*n^8-2726034480616051467*n^7+ 4633309960045658727*n^6-5949612399085521619*n^5+5682064097238550343*n^4-\ 3913859601722074602*n^3+1838192079866743314*n^2-526952131079799780*n+ 69570352728653760)/(2*n-1)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 9, b(2) = 217, b(3) = 6570 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 22 21 a(n) = 1/12 (1639513016779632768 n - 53910416451510119232 n 20 19 + 830072450788236868992 n - 7956851638960649880160 n 18 17 + 53252041472084137404456 n - 264499500376448891889892 n 16 15 + 1011741506944383210720764 n - 3052469750624653163486632 n 14 13 + 7379931977041839436468616 n - 14447065971290520540737567 n 12 11 + 23045427368838340517524412 n - 30047624432061028761258221 n 10 9 + 32030823942973132138119268 n - 27848322173621377316531387 n 8 7 + 19643658815401034030830276 n - 11147248503722087490065967 n 6 5 + 5026193678411479875215572 n - 1768927863123393877442022 n 4 3 + 473603880303164077794156 n - 92809014800222694163320 n 2 + 12506052865184969364720 n - 1032238090527948561600 n / 2 + 39233793865875840000) a(n - 1) / (n (4 n - 3) (3 n - 1) (3 n - 2) / 2 21 (4 n - 1) %1 (2 n - 1) ) + 1/2 (2 n - 3) (670693534593699456 n 20 19 - 21718377351528834816 n + 328596081210820234368 n 18 17 - 3087193851575859029664 n + 20189524303044247378296 n 16 15 - 97642750906935044479560 n + 362149002268971676100480 n 14 13 - 1054165113426369325409456 n + 2444399523838901180103332 n 12 11 - 4557001780129530541567722 n + 6863724776836985779769611 n 10 9 - 8363519788943343941603410 n + 8228568660966128691821879 n 8 7 - 6502896685926921575181112 n + 4091957495643413813726221 n 6 5 - 2023708901766177331071906 n + 772168708486260366718577 n 4 3 - 221389000949527146674090 n + 45872596076056070518820 n 2 - 6455684199018518100264 n + 550519917163258728960 n / 2 - 21456613457693568000) a(n - 2) / (n (4 n - 3) (3 n - 1) (3 n - 2) / 2 18 (4 n - 1) %1 (2 n - 1) ) + (2 n - 3) (2 n - 5) (10997332632390528 n 17 16 - 295629956641658304 n + 3650771351387965248 n 15 14 - 27481211957287202112 n + 141099108420654830184 n 13 12 - 523928388876001820308 n + 1455685020034093887228 n 11 10 - 3088845481019532568372 n + 5064726784830846944946 n 9 8 - 6451907413647681669307 n + 6386544591741674491950 n 7 6 - 4889355966791524986830 n + 2866706330281325648064 n 5 4 - 1266924355365713970311 n + 411978257627804606334 n 3 2 - 95008211263362452396 n + 14635676564635537038 n 2 / 2 - 1346881942368923580 n + 56002064767110000) (n - 2) a(n - 3) / (n / 2 (4 n - 3) (3 n - 1) (3 n - 2) (4 n - 1) %1 (2 n - 1) ) + 9 (2 n - 3) 14 13 12 (2 n - 5) (5989832588448 n - 89140503841488 n + 595774252740648 n 11 10 9 - 2367022104074924 n + 6235750221135948 n - 11502479919483460 n 8 7 6 + 15289662094565532 n - 14845167670943811 n + 10555194555981042 n 5 4 3 - 5457214243694632 n + 2016044165229840 n - 515640016794845 n 2 2 2 + 86386787952222 n - 8511914500920 n + 374445126000) (2 n - 7) (n - 2) 2 / 2 (n - 3) a(n - 4) / (n (4 n - 3) (3 n - 1) (3 n - 2) (4 n - 1) %1 / 2 (2 n - 1) ) 14 13 12 %1 := 5989832588448 n - 172998160079760 n + 2299675568228760 n 11 10 9 - 18649571498793836 n + 103084100566544896 n - 410843638546643136 n 8 7 6 + 1217598939307121952 n - 2726034480616051467 n + 4633309960045658727 n 5 4 3 - 5949612399085521619 n + 5682064097238550343 n - 3913859601722074602 n 2 + 1838192079866743314 n - 526952131079799780 n + 69570352728653760 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately -0.9321221664131375477791502027835640409617622759100764424449387320434703061\ -2009 262608102754617610852308973 10 This constant is identified as, 0 The implied delta is, -0.6587946227834138573668654075939522530992914386949486\ 941000623028593072881580365452888586212730339907 Since this is negative, there is no Apery-style irrationality proof of, 0, but still a very fast way to compute it to many digits ----------------------- This took, 2.249, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(2 n, k) binomial(2 n + 2 k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(2 n, k) binomial(2 n + 2 k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 22 21 b(n) = 1/4 (155369905048227151872 n - 5089158340315927805952 n 20 19 + 78046725411132595200000 n - 745074007490432632688640 n 18 17 + 4965709567303271473389312 n - 24560773572160705969205632 n 16 15 + 93555107887202914890911808 n - 281105341121720168563531712 n 14 13 + 676967941988222141230866496 n - 1320448870973526151851953576 n 12 11 + 2099635908686330234306643412 n - 2730586214348333186667776208 n 10 9 + 2905793907305417640087087835 n - 2524783384985815342920219829 n 8 7 + 1782307780154791671143980817 n - 1013935669217170141811079718 n 6 5 + 459251842829065219296860279 n - 162740169091882368342315791 n 4 3 + 43978405018027953967853513 n - 8719339826132035799954622 n 2 + 1191031351991560858921056 n - 99756439634181484062720 n / 2 + 3845880948066208128000) b(n - 1) / (n (4 n - 3) (3 n - 1) (3 n - 2) / 2 21 (4 n - 1) %1 (2 n - 1) ) + 1/4 (2 n - 3) (3449266344140169019392 n 20 19 - 111256465566473997336576 n + 1676059217865851067182592 n 18 17 - 15672271813909429516660224 n + 101957963852152238174548992 n 16 15 - 490249157085132848181310144 n + 1806605180538035892883579392 n 14 13 - 5221026153654765873961111792 n + 12009035379014833018727921264 n 12 11 - 22184754973639381030545086644 n + 33070753346157929862585538370 n 10 9 - 39824585143131640107134430263 n + 38655407914592598312682015697 n 8 7 - 30075352395800424201435045749 n + 18584959450485367795601560672 n 6 5 - 8998946459443513662256279439 n + 3349655484556500935160262615 n 4 3 - 932937904685841590090922537 n + 186898377430767768137881774 n 2 - 25308301417485061427675232 n + 2068615974888140183605440 n / 2 - 77203172328411308256000) b(n - 2) / (n (4 n - 3) (3 n - 1) (3 n - 2) / 2 (4 n - 1) %1 (2 n - 1) ) + 1/4 (2 n - 3) (2 n - 5) ( 18 17 60426671599266693120 n - 1616722249372675891200 n 16 15 + 19857872573948854262016 n - 148565640600510193233792 n 14 13 + 757468839206344398755520 n - 2790197388293016149196224 n 12 11 + 7681508128485838270083008 n - 16128687975193399469118344 n 10 9 + 26126741066443963788375900 n - 32818339666139939007428928 n 8 7 + 31959927731501632897816029 n - 24005790182460947465169889 n 6 5 + 13763938263739534687289203 n - 5925050682104017170122986 n 4 3 + 1868022407433883667580321 n - 415500151230473408174127 n 2 + 61408858026549265087563 n - 5399588808507071472390 n 2 / 2 + 214382165852085462000) (n - 2) b(n - 3) / (n (4 n - 3) (3 n - 1) / 2 343 (3 n - 2) (4 n - 1) %1 (2 n - 1) ) + --- (2 n - 5) (2 n - 3) ( 12 14 13 12 1260464572366848 n - 18598293495650304 n + 123085532660073024 n 11 10 - 483519442331890400 n + 1257304364926960224 n 9 8 7 - 2284574529366932728 n + 2984212691721284784 n - 2839104913685979864 n 6 5 4 + 1971119481111100077 n - 990929459254264747 n + 354188310910325322 n 3 2 - 87156871291560707 n + 13968781532813721 n - 1311073649114130 n 2 2 2 / 2 + 54930937050000) (2 n - 7) (n - 2) (n - 3) b(n - 4) / (n (4 n - 3) / 2 (3 n - 1) (3 n - 2) (4 n - 1) %1 (2 n - 1) ) 14 13 12 %1 := 1260464572366848 n - 36244797508786176 n + 479565624188910144 n 11 10 - 3870021831255023072 n + 21280500362837776000 n 9 8 - 84351234615374969304 n + 248553305368954538088 n 7 6 - 553119757183101674232 n + 934157847631187334453 n 5 4 - 1191581028958437461185 n + 1130077755080831712188 n 3 2 - 772738407517231831077 n + 360161777757167834247 n - 102425651597341043802 n + 13410389141447366880 and in Maple notation b(n) = 1/4*(155369905048227151872*n^22-5089158340315927805952*n^21+ 78046725411132595200000*n^20-745074007490432632688640*n^19+ 4965709567303271473389312*n^18-24560773572160705969205632*n^17+ 93555107887202914890911808*n^16-281105341121720168563531712*n^15+ 676967941988222141230866496*n^14-1320448870973526151851953576*n^13+ 2099635908686330234306643412*n^12-2730586214348333186667776208*n^11+ 2905793907305417640087087835*n^10-2524783384985815342920219829*n^9+ 1782307780154791671143980817*n^8-1013935669217170141811079718*n^7+ 459251842829065219296860279*n^6-162740169091882368342315791*n^5+ 43978405018027953967853513*n^4-8719339826132035799954622*n^3+ 1191031351991560858921056*n^2-99756439634181484062720*n+3845880948066208128000) /n^2/(4*n-3)/(3*n-1)/(3*n-2)/(4*n-1)/(1260464572366848*n^14-36244797508786176*n ^13+479565624188910144*n^12-3870021831255023072*n^11+21280500362837776000*n^10-\ 84351234615374969304*n^9+248553305368954538088*n^8-553119757183101674232*n^7+ 934157847631187334453*n^6-1191581028958437461185*n^5+1130077755080831712188*n^4 -772738407517231831077*n^3+360161777757167834247*n^2-102425651597341043802*n+ 13410389141447366880)/(2*n-1)^2*b(n-1)+1/4*(2*n-3)*(3449266344140169019392*n^21 -111256465566473997336576*n^20+1676059217865851067182592*n^19-\ 15672271813909429516660224*n^18+101957963852152238174548992*n^17-\ 490249157085132848181310144*n^16+1806605180538035892883579392*n^15-\ 5221026153654765873961111792*n^14+12009035379014833018727921264*n^13-\ 22184754973639381030545086644*n^12+33070753346157929862585538370*n^11-\ 39824585143131640107134430263*n^10+38655407914592598312682015697*n^9-\ 30075352395800424201435045749*n^8+18584959450485367795601560672*n^7-\ 8998946459443513662256279439*n^6+3349655484556500935160262615*n^5-\ 932937904685841590090922537*n^4+186898377430767768137881774*n^3-\ 25308301417485061427675232*n^2+2068615974888140183605440*n-\ 77203172328411308256000)/n^2/(4*n-3)/(3*n-1)/(3*n-2)/(4*n-1)/(1260464572366848* n^14-36244797508786176*n^13+479565624188910144*n^12-3870021831255023072*n^11+ 21280500362837776000*n^10-84351234615374969304*n^9+248553305368954538088*n^8-\ 553119757183101674232*n^7+934157847631187334453*n^6-1191581028958437461185*n^5+ 1130077755080831712188*n^4-772738407517231831077*n^3+360161777757167834247*n^2-\ 102425651597341043802*n+13410389141447366880)/(2*n-1)^2*b(n-2)+1/4*(2*n-3)*(2*n -5)*(60426671599266693120*n^18-1616722249372675891200*n^17+ 19857872573948854262016*n^16-148565640600510193233792*n^15+ 757468839206344398755520*n^14-2790197388293016149196224*n^13+ 7681508128485838270083008*n^12-16128687975193399469118344*n^11+ 26126741066443963788375900*n^10-32818339666139939007428928*n^9+ 31959927731501632897816029*n^8-24005790182460947465169889*n^7+ 13763938263739534687289203*n^6-5925050682104017170122986*n^5+ 1868022407433883667580321*n^4-415500151230473408174127*n^3+ 61408858026549265087563*n^2-5399588808507071472390*n+214382165852085462000)*(n-\ 2)^2/n^2/(4*n-3)/(3*n-1)/(3*n-2)/(4*n-1)/(1260464572366848*n^14-\ 36244797508786176*n^13+479565624188910144*n^12-3870021831255023072*n^11+ 21280500362837776000*n^10-84351234615374969304*n^9+248553305368954538088*n^8-\ 553119757183101674232*n^7+934157847631187334453*n^6-1191581028958437461185*n^5+ 1130077755080831712188*n^4-772738407517231831077*n^3+360161777757167834247*n^2-\ 102425651597341043802*n+13410389141447366880)/(2*n-1)^2*b(n-3)+343/12*(2*n-5)*( 2*n-3)*(1260464572366848*n^14-18598293495650304*n^13+123085532660073024*n^12-\ 483519442331890400*n^11+1257304364926960224*n^10-2284574529366932728*n^9+ 2984212691721284784*n^8-2839104913685979864*n^7+1971119481111100077*n^6-\ 990929459254264747*n^5+354188310910325322*n^4-87156871291560707*n^3+ 13968781532813721*n^2-1311073649114130*n+54930937050000)*(2*n-7)^2*(n-2)^2*(n-3 )^2/n^2/(4*n-3)/(3*n-1)/(3*n-2)/(4*n-1)/(1260464572366848*n^14-\ 36244797508786176*n^13+479565624188910144*n^12-3870021831255023072*n^11+ 21280500362837776000*n^10-84351234615374969304*n^9+248553305368954538088*n^8-\ 553119757183101674232*n^7+934157847631187334453*n^6-1191581028958437461185*n^5+ 1130077755080831712188*n^4-772738407517231831077*n^3+360161777757167834247*n^2-\ 102425651597341043802*n+13410389141447366880)/(2*n-1)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 17, b(2) = 769, b(3) = 43589 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 22 21 a(n) = 1/4 (155369905048227151872 n - 5089158340315927805952 n 20 19 + 78046725411132595200000 n - 745074007490432632688640 n 18 17 + 4965709567303271473389312 n - 24560773572160705969205632 n 16 15 + 93555107887202914890911808 n - 281105341121720168563531712 n 14 13 + 676967941988222141230866496 n - 1320448870973526151851953576 n 12 11 + 2099635908686330234306643412 n - 2730586214348333186667776208 n 10 9 + 2905793907305417640087087835 n - 2524783384985815342920219829 n 8 7 + 1782307780154791671143980817 n - 1013935669217170141811079718 n 6 5 + 459251842829065219296860279 n - 162740169091882368342315791 n 4 3 + 43978405018027953967853513 n - 8719339826132035799954622 n 2 + 1191031351991560858921056 n - 99756439634181484062720 n / 2 + 3845880948066208128000) a(n - 1) / (n (4 n - 3) (3 n - 1) (3 n - 2) / 2 21 (4 n - 1) %1 (2 n - 1) ) + 1/4 (2 n - 3) (3449266344140169019392 n 20 19 - 111256465566473997336576 n + 1676059217865851067182592 n 18 17 - 15672271813909429516660224 n + 101957963852152238174548992 n 16 15 - 490249157085132848181310144 n + 1806605180538035892883579392 n 14 13 - 5221026153654765873961111792 n + 12009035379014833018727921264 n 12 11 - 22184754973639381030545086644 n + 33070753346157929862585538370 n 10 9 - 39824585143131640107134430263 n + 38655407914592598312682015697 n 8 7 - 30075352395800424201435045749 n + 18584959450485367795601560672 n 6 5 - 8998946459443513662256279439 n + 3349655484556500935160262615 n 4 3 - 932937904685841590090922537 n + 186898377430767768137881774 n 2 - 25308301417485061427675232 n + 2068615974888140183605440 n / 2 - 77203172328411308256000) a(n - 2) / (n (4 n - 3) (3 n - 1) (3 n - 2) / 2 (4 n - 1) %1 (2 n - 1) ) + 1/4 (2 n - 3) (2 n - 5) ( 18 17 60426671599266693120 n - 1616722249372675891200 n 16 15 + 19857872573948854262016 n - 148565640600510193233792 n 14 13 + 757468839206344398755520 n - 2790197388293016149196224 n 12 11 + 7681508128485838270083008 n - 16128687975193399469118344 n 10 9 + 26126741066443963788375900 n - 32818339666139939007428928 n 8 7 + 31959927731501632897816029 n - 24005790182460947465169889 n 6 5 + 13763938263739534687289203 n - 5925050682104017170122986 n 4 3 + 1868022407433883667580321 n - 415500151230473408174127 n 2 + 61408858026549265087563 n - 5399588808507071472390 n 2 / 2 + 214382165852085462000) (n - 2) a(n - 3) / (n (4 n - 3) (3 n - 1) / 2 343 (3 n - 2) (4 n - 1) %1 (2 n - 1) ) + --- (2 n - 5) (2 n - 3) ( 12 14 13 12 1260464572366848 n - 18598293495650304 n + 123085532660073024 n 11 10 - 483519442331890400 n + 1257304364926960224 n 9 8 7 - 2284574529366932728 n + 2984212691721284784 n - 2839104913685979864 n 6 5 4 + 1971119481111100077 n - 990929459254264747 n + 354188310910325322 n 3 2 - 87156871291560707 n + 13968781532813721 n - 1311073649114130 n 2 2 2 / 2 + 54930937050000) (2 n - 7) (n - 2) (n - 3) a(n - 4) / (n (4 n - 3) / 2 (3 n - 1) (3 n - 2) (4 n - 1) %1 (2 n - 1) ) 14 13 12 %1 := 1260464572366848 n - 36244797508786176 n + 479565624188910144 n 11 10 - 3870021831255023072 n + 21280500362837776000 n 9 8 - 84351234615374969304 n + 248553305368954538088 n 7 6 - 553119757183101674232 n + 934157847631187334453 n 5 4 - 1191581028958437461185 n + 1130077755080831712188 n 3 2 - 772738407517231831077 n + 360161777757167834247 n - 102425651597341043802 n + 13410389141447366880 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.03353937970451348271373703813507305974558871617872202842519419400774485267\ 595297010996112872563316230 This constant is identified as, 3/62 ln(2) The implied delta is, -0.8579489227885534049398032813798596531368463658904285\ 475186717030499888912873791039845279316236161001 Since this is negative, there is no Apery-style irrationality proof of, 3/62 ln(2), but still a very fast way to compute it to many digits ----------------------- This took, 2.368, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(2 n, k) binomial(2 n + 2 k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(2 n, k) binomial(2 n + 2 k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 22 21 b(n) = 1/108 (1773202522138592223040128 n - 57968383229470970097940800 n 20 19 + 887350263343019533619913216 n - 8456888142633929237962932064 n 18 17 + 56283876727733765752749935112 n - 278109560393144559014410568836 n 16 + 1058932888815707720474433818900 n 15 - 3183100668992380937441875509256 n 14 + 7677255208641113541703728593744 n 13 - 15019140486763404798373859483003 n 12 + 23997613092764794651370342959556 n 11 - 31435161897450627453521248241825 n 10 + 33794565042858297182576753524924 n 9 - 29770539630405502463187380040071 n 8 + 21397098018899382044554800194164 n 7 6 - 12452457741046333450562147983371 n + 5799305179305846117873035976052 n 5 4 - 2123726263111614035468525389710 n + 595779869185627487035029176124 n 3 2 - 123031857893879428466892291864 n + 17529038371951857657287020080 n - 1529540763531984398189131200 n + 61164659259102399960960000) b(n - 1) / 2 2 / (n (4 n - 3) (3 n - 1) (3 n - 2) (4 n - 1) %1 (2 n - 1) ) + 1/18 / 21 20 (2 n - 3) (30259698266906326074819456 n - 974100520011648253216591872 n 19 18 + 14642526759952397444831748480 n - 136584303069243712851959905248 n 17 16 + 886155394517793048834614455656 n - 4247989178691186268353778021768 n 15 + 15600598894277331233878611851104 n 14 - 44910623758824651837181891313800 n 13 + 102844961970576342040207311272068 n 12 - 189030088283080695815622606640938 n 11 + 280144601672964417743740822768799 n 10 - 335070301686498614803334562958530 n 9 + 322647379911762224561098019583115 n 8 - 248668638218303614924866255820688 n 7 + 151934618964420348980407511839097 n 6 - 72568166810799071926515478141482 n 5 4 + 26564860267833594568999432043005 n - 7248651801037048460687141137634 n 3 2 + 1415968703080770455911648534420 n - 185927795943276484548438053640 n + 14652571406718771356195577600 n - 525191635382693403358080000) b(n - 2) / 2 2 / (n (4 n - 3) (3 n - 1) (3 n - 2) (4 n - 1) %1 (2 n - 1) ) + 1/9 / 18 (2 n - 3) (2 n - 5) (180101564931928329376128 n 17 16 - 4807153799245961441284032 n + 58883933773094347527314880 n 15 14 - 439152860714772227893817984 n + 2230927533690048272526574888 n 13 12 - 8183331164817549768520253252 n + 22419130077139891605177862156 n 11 10 - 46804703028238380346930764828 n + 75310626117753087284698375058 n 9 8 - 93848684664415700238198661727 n + 90528074034717438503367828518 n 7 6 - 67221177324360101270248707070 n + 38006234848095368197742784584 n 5 4 - 16081404118748913572910517611 n + 4962915567782881865188321318 n 3 2 - 1074969937225421035752299660 n + 153767358522183936619842294 n 2 - 13003982416281808791728460 n + 494581794236591351598000) (n - 2) / 2 2 b(n - 3) / (n (4 n - 3) (3 n - 1) (3 n - 2) (4 n - 1) %1 (2 n - 1) ) + / 1331 14 13 ---- (2 n - 3) (2 n - 5) (639029665947318048 n - 9388208107065177936 n 27 12 11 + 61818971391460118568 n - 241412143113034925932 n 10 9 + 623394337356699491844 n - 1123435837985043239228 n 8 7 + 1453107080687995223652 n - 1366126413770221539495 n 6 5 + 934817253445320523170 n - 461619297950334419264 n 4 3 + 161357471792728350120 n - 38614124995148396137 n 2 + 5978770560190065750 n - 538466543363273976 n + 21561488819838000) 2 2 2 / 2 (2 n - 7) (n - 2) (n - 3) b(n - 4) / (n (4 n - 3) (3 n - 1) (3 n - 2) / 2 (4 n - 1) %1 (2 n - 1) ) 14 13 %1 := 639029665947318048 n - 18334623430327630608 n 12 11 + 242017376384513374104 n - 1948126830566463997228 n 10 9 + 10683676237670357758328 n - 42227126976668318125224 n 8 7 + 124051799176661063081400 n - 275171941204132659304527 n 6 5 + 463148432527246189718019 n - 588636459311606326921775 n 4 3 + 556107994882603121100931 n - 378711745571406327498834 n 2 + 175748657969442288305034 n - 49751730089227609240788 n + 6482268968853371901120 and in Maple notation b(n) = 1/108*(1773202522138592223040128*n^22-57968383229470970097940800*n^21+ 887350263343019533619913216*n^20-8456888142633929237962932064*n^19+ 56283876727733765752749935112*n^18-278109560393144559014410568836*n^17+ 1058932888815707720474433818900*n^16-3183100668992380937441875509256*n^15+ 7677255208641113541703728593744*n^14-15019140486763404798373859483003*n^13+ 23997613092764794651370342959556*n^12-31435161897450627453521248241825*n^11+ 33794565042858297182576753524924*n^10-29770539630405502463187380040071*n^9+ 21397098018899382044554800194164*n^8-12452457741046333450562147983371*n^7+ 5799305179305846117873035976052*n^6-2123726263111614035468525389710*n^5+ 595779869185627487035029176124*n^4-123031857893879428466892291864*n^3+ 17529038371951857657287020080*n^2-1529540763531984398189131200*n+ 61164659259102399960960000)/n^2/(4*n-3)/(3*n-1)/(3*n-2)/(4*n-1)/( 639029665947318048*n^14-18334623430327630608*n^13+242017376384513374104*n^12-\ 1948126830566463997228*n^11+10683676237670357758328*n^10-\ 42227126976668318125224*n^9+124051799176661063081400*n^8-\ 275171941204132659304527*n^7+463148432527246189718019*n^6-\ 588636459311606326921775*n^5+556107994882603121100931*n^4-\ 378711745571406327498834*n^3+175748657969442288305034*n^2-\ 49751730089227609240788*n+6482268968853371901120)/(2*n-1)^2*b(n-1)+1/18*(2*n-3) *(30259698266906326074819456*n^21-974100520011648253216591872*n^20+ 14642526759952397444831748480*n^19-136584303069243712851959905248*n^18+ 886155394517793048834614455656*n^17-4247989178691186268353778021768*n^16+ 15600598894277331233878611851104*n^15-44910623758824651837181891313800*n^14+ 102844961970576342040207311272068*n^13-189030088283080695815622606640938*n^12+ 280144601672964417743740822768799*n^11-335070301686498614803334562958530*n^10+ 322647379911762224561098019583115*n^9-248668638218303614924866255820688*n^8+ 151934618964420348980407511839097*n^7-72568166810799071926515478141482*n^6+ 26564860267833594568999432043005*n^5-7248651801037048460687141137634*n^4+ 1415968703080770455911648534420*n^3-185927795943276484548438053640*n^2+ 14652571406718771356195577600*n-525191635382693403358080000)/n^2/(4*n-3)/(3*n-1 )/(3*n-2)/(4*n-1)/(639029665947318048*n^14-18334623430327630608*n^13+ 242017376384513374104*n^12-1948126830566463997228*n^11+10683676237670357758328* n^10-42227126976668318125224*n^9+124051799176661063081400*n^8-\ 275171941204132659304527*n^7+463148432527246189718019*n^6-\ 588636459311606326921775*n^5+556107994882603121100931*n^4-\ 378711745571406327498834*n^3+175748657969442288305034*n^2-\ 49751730089227609240788*n+6482268968853371901120)/(2*n-1)^2*b(n-2)+1/9*(2*n-3)* (2*n-5)*(180101564931928329376128*n^18-4807153799245961441284032*n^17+ 58883933773094347527314880*n^16-439152860714772227893817984*n^15+ 2230927533690048272526574888*n^14-8183331164817549768520253252*n^13+ 22419130077139891605177862156*n^12-46804703028238380346930764828*n^11+ 75310626117753087284698375058*n^10-93848684664415700238198661727*n^9+ 90528074034717438503367828518*n^8-67221177324360101270248707070*n^7+ 38006234848095368197742784584*n^6-16081404118748913572910517611*n^5+ 4962915567782881865188321318*n^4-1074969937225421035752299660*n^3+ 153767358522183936619842294*n^2-13003982416281808791728460*n+ 494581794236591351598000)*(n-2)^2/n^2/(4*n-3)/(3*n-1)/(3*n-2)/(4*n-1)/( 639029665947318048*n^14-18334623430327630608*n^13+242017376384513374104*n^12-\ 1948126830566463997228*n^11+10683676237670357758328*n^10-\ 42227126976668318125224*n^9+124051799176661063081400*n^8-\ 275171941204132659304527*n^7+463148432527246189718019*n^6-\ 588636459311606326921775*n^5+556107994882603121100931*n^4-\ 378711745571406327498834*n^3+175748657969442288305034*n^2-\ 49751730089227609240788*n+6482268968853371901120)/(2*n-1)^2*b(n-3)+1331/27*(2*n -3)*(2*n-5)*(639029665947318048*n^14-9388208107065177936*n^13+ 61818971391460118568*n^12-241412143113034925932*n^11+623394337356699491844*n^10 -1123435837985043239228*n^9+1453107080687995223652*n^8-1366126413770221539495*n ^7+934817253445320523170*n^6-461619297950334419264*n^5+161357471792728350120*n^ 4-38614124995148396137*n^3+5978770560190065750*n^2-538466543363273976*n+ 21561488819838000)*(2*n-7)^2*(n-2)^2*(n-3)^2/n^2/(4*n-3)/(3*n-1)/(3*n-2)/(4*n-1 )/(639029665947318048*n^14-18334623430327630608*n^13+242017376384513374104*n^12 -1948126830566463997228*n^11+10683676237670357758328*n^10-\ 42227126976668318125224*n^9+124051799176661063081400*n^8-\ 275171941204132659304527*n^7+463148432527246189718019*n^6-\ 588636459311606326921775*n^5+556107994882603121100931*n^4-\ 378711745571406327498834*n^3+175748657969442288305034*n^2-\ 49751730089227609240788*n+6482268968853371901120)/(2*n-1)^2*b(n-4) Of course, the initial conditions are b(0) = 1, b(1) = 25, b(2) = 1657, b(3) = 137458 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 22 21 a(n) = 1/108 (1773202522138592223040128 n - 57968383229470970097940800 n 20 19 + 887350263343019533619913216 n - 8456888142633929237962932064 n 18 17 + 56283876727733765752749935112 n - 278109560393144559014410568836 n 16 + 1058932888815707720474433818900 n 15 - 3183100668992380937441875509256 n 14 + 7677255208641113541703728593744 n 13 - 15019140486763404798373859483003 n 12 + 23997613092764794651370342959556 n 11 - 31435161897450627453521248241825 n 10 + 33794565042858297182576753524924 n 9 - 29770539630405502463187380040071 n 8 + 21397098018899382044554800194164 n 7 6 - 12452457741046333450562147983371 n + 5799305179305846117873035976052 n 5 4 - 2123726263111614035468525389710 n + 595779869185627487035029176124 n 3 2 - 123031857893879428466892291864 n + 17529038371951857657287020080 n - 1529540763531984398189131200 n + 61164659259102399960960000) a(n - 1) / 2 2 / (n (4 n - 3) (3 n - 1) (3 n - 2) (4 n - 1) %1 (2 n - 1) ) + 1/18 / 21 20 (2 n - 3) (30259698266906326074819456 n - 974100520011648253216591872 n 19 18 + 14642526759952397444831748480 n - 136584303069243712851959905248 n 17 16 + 886155394517793048834614455656 n - 4247989178691186268353778021768 n 15 + 15600598894277331233878611851104 n 14 - 44910623758824651837181891313800 n 13 + 102844961970576342040207311272068 n 12 - 189030088283080695815622606640938 n 11 + 280144601672964417743740822768799 n 10 - 335070301686498614803334562958530 n 9 + 322647379911762224561098019583115 n 8 - 248668638218303614924866255820688 n 7 + 151934618964420348980407511839097 n 6 - 72568166810799071926515478141482 n 5 4 + 26564860267833594568999432043005 n - 7248651801037048460687141137634 n 3 2 + 1415968703080770455911648534420 n - 185927795943276484548438053640 n + 14652571406718771356195577600 n - 525191635382693403358080000) a(n - 2) / 2 2 / (n (4 n - 3) (3 n - 1) (3 n - 2) (4 n - 1) %1 (2 n - 1) ) + 1/9 / 18 (2 n - 3) (2 n - 5) (180101564931928329376128 n 17 16 - 4807153799245961441284032 n + 58883933773094347527314880 n 15 14 - 439152860714772227893817984 n + 2230927533690048272526574888 n 13 12 - 8183331164817549768520253252 n + 22419130077139891605177862156 n 11 10 - 46804703028238380346930764828 n + 75310626117753087284698375058 n 9 8 - 93848684664415700238198661727 n + 90528074034717438503367828518 n 7 6 - 67221177324360101270248707070 n + 38006234848095368197742784584 n 5 4 - 16081404118748913572910517611 n + 4962915567782881865188321318 n 3 2 - 1074969937225421035752299660 n + 153767358522183936619842294 n 2 - 13003982416281808791728460 n + 494581794236591351598000) (n - 2) / 2 2 a(n - 3) / (n (4 n - 3) (3 n - 1) (3 n - 2) (4 n - 1) %1 (2 n - 1) ) + / 1331 14 13 ---- (2 n - 3) (2 n - 5) (639029665947318048 n - 9388208107065177936 n 27 12 11 + 61818971391460118568 n - 241412143113034925932 n 10 9 + 623394337356699491844 n - 1123435837985043239228 n 8 7 + 1453107080687995223652 n - 1366126413770221539495 n 6 5 + 934817253445320523170 n - 461619297950334419264 n 4 3 + 161357471792728350120 n - 38614124995148396137 n 2 + 5978770560190065750 n - 538466543363273976 n + 21561488819838000) 2 2 2 / 2 (2 n - 7) (n - 2) (n - 3) a(n - 4) / (n (4 n - 3) (3 n - 1) (3 n - 2) / 2 (4 n - 1) %1 (2 n - 1) ) 14 13 %1 := 639029665947318048 n - 18334623430327630608 n 12 11 + 242017376384513374104 n - 1948126830566463997228 n 10 9 + 10683676237670357758328 n - 42227126976668318125224 n 8 7 + 124051799176661063081400 n - 275171941204132659304527 n 6 5 + 463148432527246189718019 n - 588636459311606326921775 n 4 3 + 556107994882603121100931 n - 378711745571406327498834 n 2 + 175748657969442288305034 n - 49751730089227609240788 n + 6482268968853371901120 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.01979481601203801245757198625085631900265748752833782795918368168716205933\ 727223363736244603267985746 This constant is identified as, 2/111 ln(3) The implied delta is, -0.9401697853158025989823513299282604463715072866221295\ 064358737361301365053729048176045488687310363187 Since this is negative, there is no Apery-style irrationality proof of, 2/111 ln(3), but still a very fast way to compute it to many digits ----------------------- This took, 2.259, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ ) binomial(n, k) binomial(2 n + k, k) binomial(2 n + 2 k, k) / ----- k By Shalosh B. Ekhad Let ----- \ b(n) = ) binomial(n, k) binomial(2 n + k, k) binomial(2 n + 2 k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 7 6 5 4 3 b(n) = 1/5 (105090 n - 928521 n + 3299409 n - 6053082 n + 6126321 n 2 / 2 - 3376667 n + 924275 n - 94875) b(n - 1) / ((2 n - 1) %1 n ) + 1/5 ( / 8 7 6 5 4 3 17980 n - 203812 n + 977389 n - 2573355 n + 4031675 n - 3797150 n 2 / 2 + 2056516 n - 567425 n + 58875) b(n - 2) / (n (2 n - 1) (2 n - 3) %1) / 4 3 2 2 2 (155 n - 517 n + 574 n - 235 n + 30) (n - 2) (2 n - 5) b(n - 3) + 1/5 -------------------------------------------------------------------- 2 n (2 n - 1) (2 n - 3) %1 4 3 2 %1 := 155 n - 1137 n + 3055 n - 3554 n + 1511 and in Maple notation b(n) = 1/5*(105090*n^7-928521*n^6+3299409*n^5-6053082*n^4+6126321*n^3-3376667*n ^2+924275*n-94875)/(2*n-1)/(155*n^4-1137*n^3+3055*n^2-3554*n+1511)/n^2*b(n-1)+1 /5*(17980*n^8-203812*n^7+977389*n^6-2573355*n^5+4031675*n^4-3797150*n^3+2056516 *n^2-567425*n+58875)/n^2/(2*n-1)/(2*n-3)/(155*n^4-1137*n^3+3055*n^2-3554*n+1511 )*b(n-2)+1/5*(155*n^4-517*n^3+574*n^2-235*n+30)*(n-2)^2*(2*n-5)^2/n^2/(2*n-1)/( 2*n-3)/(155*n^4-1137*n^3+3055*n^2-3554*n+1511)*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 13, b(2) = 481 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 7 6 5 4 3 a(n) = 1/5 (105090 n - 928521 n + 3299409 n - 6053082 n + 6126321 n 2 / 2 - 3376667 n + 924275 n - 94875) a(n - 1) / ((2 n - 1) %1 n ) + 1/5 ( / 8 7 6 5 4 3 17980 n - 203812 n + 977389 n - 2573355 n + 4031675 n - 3797150 n 2 / 2 + 2056516 n - 567425 n + 58875) a(n - 2) / (n (2 n - 1) (2 n - 3) %1) / 4 3 2 2 2 (155 n - 517 n + 574 n - 235 n + 30) (n - 2) (2 n - 5) a(n - 3) + 1/5 -------------------------------------------------------------------- 2 n (2 n - 1) (2 n - 3) %1 4 3 2 %1 := 155 n - 1137 n + 3055 n - 3554 n + 1511 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.07685873945785786408739879159832239739724476473933743834946537571295735990\ 377356452883754726736070632 The implied delta is, -0.0855134697182789664679060265268265421399827307430878\ 568051287775347370551492004742090097977649003283 Since this is negative, there is no Apery-style irrationality proof of, 0.076\ 858739457857864087398791598322397397244764739337438349465375712957359903\ 77356452883754726736070632, but still a very fast way to compute it to many digits ----------------------- This took, 3.989, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(2 n + k, k) binomial(2 n + 2 k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(2 n + k, k) binomial(2 n + 2 k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 7 6 5 4 3 b(n) = 1/9 (671842 n - 5929065 n + 21044261 n - 38570276 n + 39014381 n 2 / 2 - 21504891 n + 5892183 n - 606285) b(n - 1) / ((2 n - 1) %1 n ) + 1/9 ( / 8 7 6 5 4 3 69052 n - 782020 n + 3744681 n - 9838047 n + 15366957 n - 14414302 n 2 / 2 + 7765198 n - 2129013 n + 220185) b(n - 2) / (n (2 n - 1) (2 n - 3) %1 / ) + 4 3 2 2 2 (283 n - 941 n + 1040 n - 423 n + 54) (n - 2) (2 n - 5) b(n - 3) 1/9 --------------------------------------------------------------------- 2 n (2 n - 1) (2 n - 3) %1 4 3 2 %1 := 283 n - 2073 n + 5561 n - 6458 n + 2741 and in Maple notation b(n) = 1/9*(671842*n^7-5929065*n^6+21044261*n^5-38570276*n^4+39014381*n^3-\ 21504891*n^2+5892183*n-606285)/(2*n-1)/(283*n^4-2073*n^3+5561*n^2-6458*n+2741)/ n^2*b(n-1)+1/9*(69052*n^8-782020*n^7+3744681*n^6-9838047*n^5+15366957*n^4-\ 14414302*n^3+7765198*n^2-2129013*n+220185)/n^2/(2*n-1)/(2*n-3)/(283*n^4-2073*n^ 3+5561*n^2-6458*n+2741)*b(n-2)+1/9*(283*n^4-941*n^3+1040*n^2-423*n+54)*(n-2)^2* (2*n-5)^2/n^2/(2*n-1)/(2*n-3)/(283*n^4-2073*n^3+5561*n^2-6458*n+2741)*b(n-3) Of course, the initial conditions are b(0) = 1, b(1) = 25, b(2) = 1801 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 7 6 5 4 3 a(n) = 1/9 (671842 n - 5929065 n + 21044261 n - 38570276 n + 39014381 n 2 / 2 - 21504891 n + 5892183 n - 606285) a(n - 1) / ((2 n - 1) %1 n ) + 1/9 ( / 8 7 6 5 4 3 69052 n - 782020 n + 3744681 n - 9838047 n + 15366957 n - 14414302 n 2 / 2 + 7765198 n - 2129013 n + 220185) a(n - 2) / (n (2 n - 1) (2 n - 3) %1 / ) + 4 3 2 2 2 (283 n - 941 n + 1040 n - 423 n + 54) (n - 2) (2 n - 5) a(n - 3) 1/9 --------------------------------------------------------------------- 2 n (2 n - 1) (2 n - 3) %1 4 3 2 %1 := 283 n - 2073 n + 5561 n - 6458 n + 2741 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.03998926041691992169714800700720249431204808467463011081465461593231117049\ 825161820803057655748569351 This constant is identified as, 3/52 ln(2) The implied delta is, -0.0698752982845850019438738883822275673980536310869105\ 366290884748929148133214400753376868128909392394 Since this is negative, there is no Apery-style irrationality proof of, 3/52 ln(2), but still a very fast way to compute it to many digits ----------------------- This took, 1.374, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ k ) binomial(n, k) binomial(2 n + k, k) binomial(2 n + 2 k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ k b(n) = ) binomial(n, k) binomial(2 n + k, k) binomial(2 n + 2 k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 7 6 5 4 b(n) = 1/13 (2093634 n - 18468297 n + 65521593 n - 120043470 n 3 2 / + 121395945 n - 66912347 n + 18339035 n - 1888575) b(n - 1) / ( / 2 8 7 6 5 (2 n - 1) %1 n ) + 1/13 (152892 n - 1730916 n + 8284005 n - 21747027 n 4 3 2 + 33932559 n - 31783294 n + 17089928 n - 4675177 n + 483015) b(n - 2) / 2 / (n (2 n - 1) (2 n - 3) %1) + / 4 3 2 2 2 (411 n - 1365 n + 1506 n - 611 n + 78) (n - 2) (2 n - 5) b(n - 3) 1/13 ---------------------------------------------------------------------- 2 n (2 n - 1) (2 n - 3) %1 4 3 2 %1 := 411 n - 3009 n + 8067 n - 9362 n + 3971 and in Maple notation b(n) = 1/13*(2093634*n^7-18468297*n^6+65521593*n^5-120043470*n^4+121395945*n^3-\ 66912347*n^2+18339035*n-1888575)/(2*n-1)/(411*n^4-3009*n^3+8067*n^2-9362*n+3971 )/n^2*b(n-1)+1/13*(152892*n^8-1730916*n^7+8284005*n^6-21747027*n^5+33932559*n^4 -31783294*n^3+17089928*n^2-4675177*n+483015)/n^2/(2*n-1)/(2*n-3)/(411*n^4-3009* n^3+8067*n^2-9362*n+3971)*b(n-2)+1/13*(411*n^4-1365*n^3+1506*n^2-611*n+78)*(n-2 )^2*(2*n-5)^2/n^2/(2*n-1)/(2*n-3)/(411*n^4-3009*n^3+8067*n^2-9362*n+3971)*b(n-3 ) Of course, the initial conditions are b(0) = 1, b(1) = 37, b(2) = 3961 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 7 6 5 4 a(n) = 1/13 (2093634 n - 18468297 n + 65521593 n - 120043470 n 3 2 / + 121395945 n - 66912347 n + 18339035 n - 1888575) a(n - 1) / ( / 2 8 7 6 5 (2 n - 1) %1 n ) + 1/13 (152892 n - 1730916 n + 8284005 n - 21747027 n 4 3 2 + 33932559 n - 31783294 n + 17089928 n - 4675177 n + 483015) a(n - 2) / 2 / (n (2 n - 1) (2 n - 3) %1) + / 4 3 2 2 2 (411 n - 1365 n + 1506 n - 611 n + 78) (n - 2) (2 n - 5) a(n - 3) 1/13 ---------------------------------------------------------------------- 2 n (2 n - 1) (2 n - 3) %1 4 3 2 %1 := 411 n - 3009 n + 8067 n - 9362 n + 3971 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.02702351667256146460142264421639296119425563606429443859521418350951418291\ 647948790030521301167253317 The implied delta is, -0.0628187094416363735434321807538920273377947445279044\ 102488017395998266473362360991745629260809079320 Since this is negative, there is no Apery-style irrationality proof of, 0.027\ 023516672561464601422644216392961194255636064294438595214183509514182916\ 47948790030521301167253317, but still a very fast way to compute it to many digits ----------------------- This took, 3.973, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 ) binomial(n, k) binomial(2 n + 2 k, k) / ----- k By Shalosh B. Ekhad Let ----- \ 2 b(n) = ) binomial(n, k) binomial(2 n + 2 k, k) / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 21 20 b(n) = 2/9 (12264942295820112800 n - 501257429949631124000 n 19 18 + 9552936256749875150952 n - 112823125352634537877544 n 17 16 + 925590108670722447943696 n - 5603777427681289863449826 n 15 14 + 25963870738764864457047866 n - 94219505631906320256618404 n 13 12 + 271826519384835697695340661 n - 629310684684682032832455974 n 11 10 + 1175016511890562155674781845 n - 1772007305743106075382100950 n 9 8 + 2154532257339364859114282303 n - 2101392578243644510130470760 n 7 6 + 1629885172372250621765393581 n - 992273149785218622418576406 n 5 4 + 465320158071689983864688040 n - 163577097288787337692103160 n 3 2 + 41396652835612087778670096 n - 7069423086271251162634176 n / 2 + 723885597952707588053760 n - 33338030586067604275200) b(n - 1) / (n / 2 2 21 %1 (3 n - 1) (3 n - 2) ) + 2/9 (8609151174658872640 n 20 19 - 360457605718798163840 n + 7049700370223073092208 n 18 17 - 85591939684917605240248 n + 723146678409137573562768 n 16 15 - 4516850482577620807706460 n + 21628999604836123054342572 n 14 13 - 81257226514450603208058228 n + 243091514711472097911978191 n 12 11 - 584445342904994063737119648 n + 1134715215756404552067858929 n 10 9 - 1781271145837591343043944670 n + 2256124736801221039549626641 n 8 7 - 2293103061283060809829622728 n + 1853304908885893914692857147 n 6 5 - 1175006690566717008468606890 n + 573168721404261794258755480 n 4 3 - 209228342011254735351067464 n + 54854465277610805819021424 n 2 - 9675336388373583087052224 n + 1019205877451667543590400 n / 2 2 2 - 48024946341427249152000) b(n - 2) / (n %1 (3 n - 1) (3 n - 2) ) + / 20 19 4/9 (n - 2) (4828705959436102656 n - 197344975690125204480 n 18 17 + 3751135092656797021744 n - 44044375678817353634976 n 16 15 + 357838663350155364016368 n - 2135456274377577987651440 n 14 13 + 9698035961888898102281052 n - 34265218545733530300302123 n 12 11 + 95487267848009538988708275 n - 211519882777502246808052834 n 10 9 + 373661141718710246203767240 n - 526049424861934000836604567 n 8 7 + 587586754843545315277919259 n - 516385326879973779182192116 n 6 5 + 352446602353030703990052446 n - 183326222326496390811063224 n 4 3 + 70716034988333022397059792 n - 19424193268586496054302880 n 2 + 3560966839743400463064768 n - 387177359184837230290560 n / 2 2 2 + 18744839706293445427200) b(n - 3) / (n %1 (3 n - 1) (3 n - 2) ) + / 19 18 4/9 (n - 2) (n - 3) (323663261773796736 n - 12580506991913697408 n 17 16 + 226073102120964458176 n - 2492966600321067296880 n 15 14 + 18883114059464893807536 n - 104212798960691226797476 n 13 12 + 433775825702615741103308 n - 1390803928692073915069604 n 11 10 + 3478353881473714211215251 n - 6829600992192348778530406 n 9 8 + 10544547604788745849249830 n - 12766911494101820588116136 n 7 6 + 12036453863835505187276311 n - 8731409651521897146726322 n 5 4 + 4786532232366492354783428 n - 1931061937854408711618344 n 3 2 + 550921751291744746173264 n - 104246479510004286258624 n / 2 + 11633419308211385837760 n - 575259971188267008000) b(n - 4) / (n %1 / 2 2 15 (3 n - 1) (3 n - 2) ) + 68/9 (n - 2) (n - 3) (364485655150672 n 14 13 12 - 8335467121095664 n + 86196179536229520 n - 533659113296059620 n 11 10 + 2207759965834055572 n - 6450455304714687196 n 9 8 + 13716708472744243835 n - 21560107754357960204 n 7 6 + 25183185730497666502 n - 21792373119058269428 n 5 4 + 13813049599864660047 n - 6277506712121051920 n 3 2 + 1973642311043392972 n - 403980822807445568 n + 48007504158541680 n 2 2 / 2 2 - 2495958645196800) (n - 4) (2 n - 9) b(n - 5) / (n %1 (3 n - 1) / 2 (3 n - 2) ) 15 14 13 %1 := 364485655150672 n - 13802751948355744 n + 241163713022389376 n 12 11 - 2578577928380304564 n + 18866604280572162548 n 10 9 - 100047776764425103808 n + 397195188814764028143 n 8 7 - 1202033984184058005371 n + 2795604428240298897002 n 6 5 - 4996411243508907625770 n + 6805959987860378214071 n 4 3 - 6938941920108342354199 n + 5125634768267368940292 n 2 - 2589776352110661581952 n + 800356484237474059704 n - 114057828501455707200 and in Maple notation b(n) = 2/9*(12264942295820112800*n^21-501257429949631124000*n^20+ 9552936256749875150952*n^19-112823125352634537877544*n^18+ 925590108670722447943696*n^17-5603777427681289863449826*n^16+ 25963870738764864457047866*n^15-94219505631906320256618404*n^14+ 271826519384835697695340661*n^13-629310684684682032832455974*n^12+ 1175016511890562155674781845*n^11-1772007305743106075382100950*n^10+ 2154532257339364859114282303*n^9-2101392578243644510130470760*n^8+ 1629885172372250621765393581*n^7-992273149785218622418576406*n^6+ 465320158071689983864688040*n^5-163577097288787337692103160*n^4+ 41396652835612087778670096*n^3-7069423086271251162634176*n^2+ 723885597952707588053760*n-33338030586067604275200)/n^2/(364485655150672*n^15-\ 13802751948355744*n^14+241163713022389376*n^13-2578577928380304564*n^12+ 18866604280572162548*n^11-100047776764425103808*n^10+397195188814764028143*n^9-\ 1202033984184058005371*n^8+2795604428240298897002*n^7-4996411243508907625770*n^ 6+6805959987860378214071*n^5-6938941920108342354199*n^4+5125634768267368940292* n^3-2589776352110661581952*n^2+800356484237474059704*n-114057828501455707200)/( 3*n-1)^2/(3*n-2)^2*b(n-1)+2/9*(8609151174658872640*n^21-360457605718798163840*n ^20+7049700370223073092208*n^19-85591939684917605240248*n^18+ 723146678409137573562768*n^17-4516850482577620807706460*n^16+ 21628999604836123054342572*n^15-81257226514450603208058228*n^14+ 243091514711472097911978191*n^13-584445342904994063737119648*n^12+ 1134715215756404552067858929*n^11-1781271145837591343043944670*n^10+ 2256124736801221039549626641*n^9-2293103061283060809829622728*n^8+ 1853304908885893914692857147*n^7-1175006690566717008468606890*n^6+ 573168721404261794258755480*n^5-209228342011254735351067464*n^4+ 54854465277610805819021424*n^3-9675336388373583087052224*n^2+ 1019205877451667543590400*n-48024946341427249152000)/n^2/(364485655150672*n^15-\ 13802751948355744*n^14+241163713022389376*n^13-2578577928380304564*n^12+ 18866604280572162548*n^11-100047776764425103808*n^10+397195188814764028143*n^9-\ 1202033984184058005371*n^8+2795604428240298897002*n^7-4996411243508907625770*n^ 6+6805959987860378214071*n^5-6938941920108342354199*n^4+5125634768267368940292* n^3-2589776352110661581952*n^2+800356484237474059704*n-114057828501455707200)/( 3*n-1)^2/(3*n-2)^2*b(n-2)+4/9*(n-2)*(4828705959436102656*n^20-\ 197344975690125204480*n^19+3751135092656797021744*n^18-44044375678817353634976* n^17+357838663350155364016368*n^16-2135456274377577987651440*n^15+ 9698035961888898102281052*n^14-34265218545733530300302123*n^13+ 95487267848009538988708275*n^12-211519882777502246808052834*n^11+ 373661141718710246203767240*n^10-526049424861934000836604567*n^9+ 587586754843545315277919259*n^8-516385326879973779182192116*n^7+ 352446602353030703990052446*n^6-183326222326496390811063224*n^5+ 70716034988333022397059792*n^4-19424193268586496054302880*n^3+ 3560966839743400463064768*n^2-387177359184837230290560*n+ 18744839706293445427200)/n^2/(364485655150672*n^15-13802751948355744*n^14+ 241163713022389376*n^13-2578577928380304564*n^12+18866604280572162548*n^11-\ 100047776764425103808*n^10+397195188814764028143*n^9-1202033984184058005371*n^8 +2795604428240298897002*n^7-4996411243508907625770*n^6+6805959987860378214071*n ^5-6938941920108342354199*n^4+5125634768267368940292*n^3-2589776352110661581952 *n^2+800356484237474059704*n-114057828501455707200)/(3*n-1)^2/(3*n-2)^2*b(n-3)+ 4/9*(n-2)*(n-3)*(323663261773796736*n^19-12580506991913697408*n^18+ 226073102120964458176*n^17-2492966600321067296880*n^16+18883114059464893807536* n^15-104212798960691226797476*n^14+433775825702615741103308*n^13-\ 1390803928692073915069604*n^12+3478353881473714211215251*n^11-\ 6829600992192348778530406*n^10+10544547604788745849249830*n^9-\ 12766911494101820588116136*n^8+12036453863835505187276311*n^7-\ 8731409651521897146726322*n^6+4786532232366492354783428*n^5-\ 1931061937854408711618344*n^4+550921751291744746173264*n^3-\ 104246479510004286258624*n^2+11633419308211385837760*n-575259971188267008000)/n ^2/(364485655150672*n^15-13802751948355744*n^14+241163713022389376*n^13-\ 2578577928380304564*n^12+18866604280572162548*n^11-100047776764425103808*n^10+ 397195188814764028143*n^9-1202033984184058005371*n^8+2795604428240298897002*n^7 -4996411243508907625770*n^6+6805959987860378214071*n^5-6938941920108342354199*n ^4+5125634768267368940292*n^3-2589776352110661581952*n^2+800356484237474059704* n-114057828501455707200)/(3*n-1)^2/(3*n-2)^2*b(n-4)+68/9*(n-2)*(n-3)*( 364485655150672*n^15-8335467121095664*n^14+86196179536229520*n^13-\ 533659113296059620*n^12+2207759965834055572*n^11-6450455304714687196*n^10+ 13716708472744243835*n^9-21560107754357960204*n^8+25183185730497666502*n^7-\ 21792373119058269428*n^6+13813049599864660047*n^5-6277506712121051920*n^4+ 1973642311043392972*n^3-403980822807445568*n^2+48007504158541680*n-\ 2495958645196800)*(n-4)^2*(2*n-9)^2/n^2/(364485655150672*n^15-13802751948355744 *n^14+241163713022389376*n^13-2578577928380304564*n^12+18866604280572162548*n^ 11-100047776764425103808*n^10+397195188814764028143*n^9-1202033984184058005371* n^8+2795604428240298897002*n^7-4996411243508907625770*n^6+ 6805959987860378214071*n^5-6938941920108342354199*n^4+5125634768267368940292*n^ 3-2589776352110661581952*n^2+800356484237474059704*n-114057828501455707200)/(3* n-1)^2/(3*n-2)^2*b(n-5) Of course, the initial conditions are b(0) = 1, b(1) = 17, b(2) = 857, b(3) = 54668, b(4) = 3868921 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 21 20 a(n) = 2/9 (12264942295820112800 n - 501257429949631124000 n 19 18 + 9552936256749875150952 n - 112823125352634537877544 n 17 16 + 925590108670722447943696 n - 5603777427681289863449826 n 15 14 + 25963870738764864457047866 n - 94219505631906320256618404 n 13 12 + 271826519384835697695340661 n - 629310684684682032832455974 n 11 10 + 1175016511890562155674781845 n - 1772007305743106075382100950 n 9 8 + 2154532257339364859114282303 n - 2101392578243644510130470760 n 7 6 + 1629885172372250621765393581 n - 992273149785218622418576406 n 5 4 + 465320158071689983864688040 n - 163577097288787337692103160 n 3 2 + 41396652835612087778670096 n - 7069423086271251162634176 n / 2 + 723885597952707588053760 n - 33338030586067604275200) a(n - 1) / (n / 2 2 21 %1 (3 n - 1) (3 n - 2) ) + 2/9 (8609151174658872640 n 20 19 - 360457605718798163840 n + 7049700370223073092208 n 18 17 - 85591939684917605240248 n + 723146678409137573562768 n 16 15 - 4516850482577620807706460 n + 21628999604836123054342572 n 14 13 - 81257226514450603208058228 n + 243091514711472097911978191 n 12 11 - 584445342904994063737119648 n + 1134715215756404552067858929 n 10 9 - 1781271145837591343043944670 n + 2256124736801221039549626641 n 8 7 - 2293103061283060809829622728 n + 1853304908885893914692857147 n 6 5 - 1175006690566717008468606890 n + 573168721404261794258755480 n 4 3 - 209228342011254735351067464 n + 54854465277610805819021424 n 2 - 9675336388373583087052224 n + 1019205877451667543590400 n / 2 2 2 - 48024946341427249152000) a(n - 2) / (n %1 (3 n - 1) (3 n - 2) ) + / 20 19 4/9 (n - 2) (4828705959436102656 n - 197344975690125204480 n 18 17 + 3751135092656797021744 n - 44044375678817353634976 n 16 15 + 357838663350155364016368 n - 2135456274377577987651440 n 14 13 + 9698035961888898102281052 n - 34265218545733530300302123 n 12 11 + 95487267848009538988708275 n - 211519882777502246808052834 n 10 9 + 373661141718710246203767240 n - 526049424861934000836604567 n 8 7 + 587586754843545315277919259 n - 516385326879973779182192116 n 6 5 + 352446602353030703990052446 n - 183326222326496390811063224 n 4 3 + 70716034988333022397059792 n - 19424193268586496054302880 n 2 + 3560966839743400463064768 n - 387177359184837230290560 n / 2 2 2 + 18744839706293445427200) a(n - 3) / (n %1 (3 n - 1) (3 n - 2) ) + / 19 18 4/9 (n - 2) (n - 3) (323663261773796736 n - 12580506991913697408 n 17 16 + 226073102120964458176 n - 2492966600321067296880 n 15 14 + 18883114059464893807536 n - 104212798960691226797476 n 13 12 + 433775825702615741103308 n - 1390803928692073915069604 n 11 10 + 3478353881473714211215251 n - 6829600992192348778530406 n 9 8 + 10544547604788745849249830 n - 12766911494101820588116136 n 7 6 + 12036453863835505187276311 n - 8731409651521897146726322 n 5 4 + 4786532232366492354783428 n - 1931061937854408711618344 n 3 2 + 550921751291744746173264 n - 104246479510004286258624 n / 2 + 11633419308211385837760 n - 575259971188267008000) a(n - 4) / (n %1 / 2 2 15 (3 n - 1) (3 n - 2) ) + 68/9 (n - 2) (n - 3) (364485655150672 n 14 13 12 - 8335467121095664 n + 86196179536229520 n - 533659113296059620 n 11 10 + 2207759965834055572 n - 6450455304714687196 n 9 8 + 13716708472744243835 n - 21560107754357960204 n 7 6 + 25183185730497666502 n - 21792373119058269428 n 5 4 + 13813049599864660047 n - 6277506712121051920 n 3 2 + 1973642311043392972 n - 403980822807445568 n + 48007504158541680 n 2 2 / 2 2 - 2495958645196800) (n - 4) (2 n - 9) a(n - 5) / (n %1 (3 n - 1) / 2 (3 n - 2) ) 15 14 13 %1 := 364485655150672 n - 13802751948355744 n + 241163713022389376 n 12 11 - 2578577928380304564 n + 18866604280572162548 n 10 9 - 100047776764425103808 n + 397195188814764028143 n 8 7 - 1202033984184058005371 n + 2795604428240298897002 n 6 5 - 4996411243508907625770 n + 6805959987860378214071 n 4 3 - 6938941920108342354199 n + 5125634768267368940292 n 2 - 2589776352110661581952 n + 800356484237474059704 n - 114057828501455707200 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 0, a(4) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately -0.1796315710002647325890058588785440992672970111601588341581008657052843668\ -4070 125713152417998375722884553 10 This constant is identified as, 0 The implied delta is, -0.3776576829047592084752137195949687273230458848427419\ 646165351757539234100926190223494139276350408376 Since this is negative, there is no Apery-style irrationality proof of, 0, but still a very fast way to compute it to many digits ----------------------- This took, 2.953, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(2 n + 2 k, k) 2 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(2 n + 2 k, k) 2 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 21 20 b(n) = 1/18 (5462083988794181575168 n - 223245717281881556743424 n 19 18 + 4255004887063514446388544 n - 50258963098906439412928544 n 17 16 + 412380036293510751187713504 n - 2497092995372570094928567576 n 15 14 + 11572036430603475049748255208 n - 42002944238390782118831875114 n 13 12 + 121211009353311708102579577819 n - 280698794599642853983711044656 n 11 10 + 524276062550581932318282990315 n - 790933019434812438196740003020 n 9 8 + 962067661879164383038413522233 n - 938776069358884642417825306704 n 7 6 + 728521052293052544366044489945 n - 443792256052446436792915770122 n 5 4 + 208258243909832999896875112176 n - 73268588166745182041860584840 n 3 2 + 18558804977597965532024307888 n - 3172447797633475002711259200 n + 325170066792690929135289600 n - 14987450772753290625024000) b(n - 1) / 2 2 2 21 / (n %1 (3 n - 1) (3 n - 2) ) + 1/36 (4658855662089231956224 n / 20 19 - 195075109994617945155456 n + 3815248772647586074073280 n 18 17 - 46319126110040976546162240 n + 391286516119147423955760848 n 16 15 - 2443464565870966677944829816 n + 11696743043639881904438813464 n 14 13 - 43923920346742702717458485492 n + 131331441048677697873105114595 n 12 11 - 315536544154679255283235261567 n + 612132678688881432891158766910 n 10 9 - 960037432996574731806800688678 n + 1214700651816918316961542121935 n 8 7 - 1233190223476442593064249943139 n + 995431543843088742860488349288 n 6 5 - 630270921840367962724386035228 n + 307018090521789415948895980128 n 4 3 - 111912478526480431761073233072 n + 29298635452173767552952189408 n 2 - 5161009633717690120540830912 n + 543233520160201213208334720 n / 2 2 2 - 25614618339244074939187200) b(n - 2) / (n %1 (3 n - 1) (3 n - 2) ) / 20 19 + 1/36 (n - 2) (1568442269758218600704 n - 64105206043288088543872 n 18 17 + 1218695172261502724462272 n - 14312815398702836566154752 n 16 15 + 116322191053800465053775824 n - 694458971365097895127357464 n 14 13 + 3155452221421305861514977296 n - 11155676652688152456939335520 n 12 11 + 31109747724762666743867218529 n - 68969356669198846783434558947 n 10 9 + 121950513185622810631996977520 n - 171862613511873758878288235394 n 8 7 + 192188202682655652758367119189 n - 169115133125502420020916628271 n 6 5 + 115588544267200615669287330774 n - 60217958197238781562156538724 n 4 3 + 23269099089634169136247659204 n - 6404228497627538563660458096 n 2 + 1176760532756580208597973088 n - 128287367020776725858129760 n / 2 2 2 + 6227754420076623544320000) b(n - 3) / (n %1 (3 n - 1) (3 n - 2) ) + / 19 18 1/36 (n - 2) (n - 3) (78738215814172205312 n - 3060703571277653438592 n 17 16 + 55002876150062012824512 n - 606533084140754111731072 n 15 14 + 4594173788016315630386800 n - 25354276222810749181953688 n 13 12 + 105535594162848712186915904 n - 338389379547709667253718636 n 11 10 + 846373970688528469585900607 n - 1662069261249598740203466314 n 9 8 + 2566740758243685350539302806 n - 3108733725291158422207230632 n 7 6 + 2932195647060171908233878263 n - 2128337289970643847072107750 n 5 4 + 1167666778279191106183552708 n - 471559350462241436713422660 n 3 2 + 134710377233222887428509568 n - 25533578219638220391747456 n / 2 + 2855611262538225282172320 n - 141537579974663336755200) b(n - 4) / (n / 2 2 11 15 %1 (3 n - 1) (3 n - 2) ) + -- (n - 2) (n - 3) (20526125081900992 n 12 14 13 - 469471355122757600 n + 4855727929849116448 n 12 11 - 30071390104303927240 n + 124452090305359659580 n 10 9 - 363781641128706757576 n + 774002613580364911215 n 8 7 - 1217396791110327898100 n + 1423097041295213928482 n 6 5 - 1232639346246718854812 n + 782185650712061928167 n 4 3 - 355959158795058424160 n + 112101330751111620956 n 2 - 22993991052405656192 n + 2739618485091811440 n - 142809368673484800) 2 2 / 2 2 2 (n - 4) (2 n - 9) b(n - 5) / (n %1 (3 n - 1) (3 n - 2) ) / 15 14 13 %1 := 20526125081900992 n - 777363231351272480 n + 13583570035167327008 n 12 11 - 145257133420778334024 n + 1062961284086716489884 n 10 9 - 5637785349407398551500 n + 22386850193441871197155 n 8 7 - 67764757106191690829351 n + 157641633591197603653110 n 6 5 - 281819608004724099992254 n + 383998832909107706933795 n 4 3 - 391625455736721914272295 n + 289382480493137658782544 n 2 - 146266116223527949810896 n + 45220160739070155931272 n - 6446909198345452637760 and in Maple notation b(n) = 1/18*(5462083988794181575168*n^21-223245717281881556743424*n^20+ 4255004887063514446388544*n^19-50258963098906439412928544*n^18+ 412380036293510751187713504*n^17-2497092995372570094928567576*n^16+ 11572036430603475049748255208*n^15-42002944238390782118831875114*n^14+ 121211009353311708102579577819*n^13-280698794599642853983711044656*n^12+ 524276062550581932318282990315*n^11-790933019434812438196740003020*n^10+ 962067661879164383038413522233*n^9-938776069358884642417825306704*n^8+ 728521052293052544366044489945*n^7-443792256052446436792915770122*n^6+ 208258243909832999896875112176*n^5-73268588166745182041860584840*n^4+ 18558804977597965532024307888*n^3-3172447797633475002711259200*n^2+ 325170066792690929135289600*n-14987450772753290625024000)/n^2/( 20526125081900992*n^15-777363231351272480*n^14+13583570035167327008*n^13-\ 145257133420778334024*n^12+1062961284086716489884*n^11-5637785349407398551500*n ^10+22386850193441871197155*n^9-67764757106191690829351*n^8+ 157641633591197603653110*n^7-281819608004724099992254*n^6+ 383998832909107706933795*n^5-391625455736721914272295*n^4+ 289382480493137658782544*n^3-146266116223527949810896*n^2+ 45220160739070155931272*n-6446909198345452637760)/(3*n-1)^2/(3*n-2)^2*b(n-1)+1/ 36*(4658855662089231956224*n^21-195075109994617945155456*n^20+ 3815248772647586074073280*n^19-46319126110040976546162240*n^18+ 391286516119147423955760848*n^17-2443464565870966677944829816*n^16+ 11696743043639881904438813464*n^15-43923920346742702717458485492*n^14+ 131331441048677697873105114595*n^13-315536544154679255283235261567*n^12+ 612132678688881432891158766910*n^11-960037432996574731806800688678*n^10+ 1214700651816918316961542121935*n^9-1233190223476442593064249943139*n^8+ 995431543843088742860488349288*n^7-630270921840367962724386035228*n^6+ 307018090521789415948895980128*n^5-111912478526480431761073233072*n^4+ 29298635452173767552952189408*n^3-5161009633717690120540830912*n^2+ 543233520160201213208334720*n-25614618339244074939187200)/n^2/( 20526125081900992*n^15-777363231351272480*n^14+13583570035167327008*n^13-\ 145257133420778334024*n^12+1062961284086716489884*n^11-5637785349407398551500*n ^10+22386850193441871197155*n^9-67764757106191690829351*n^8+ 157641633591197603653110*n^7-281819608004724099992254*n^6+ 383998832909107706933795*n^5-391625455736721914272295*n^4+ 289382480493137658782544*n^3-146266116223527949810896*n^2+ 45220160739070155931272*n-6446909198345452637760)/(3*n-1)^2/(3*n-2)^2*b(n-2)+1/ 36*(n-2)*(1568442269758218600704*n^20-64105206043288088543872*n^19+ 1218695172261502724462272*n^18-14312815398702836566154752*n^17+ 116322191053800465053775824*n^16-694458971365097895127357464*n^15+ 3155452221421305861514977296*n^14-11155676652688152456939335520*n^13+ 31109747724762666743867218529*n^12-68969356669198846783434558947*n^11+ 121950513185622810631996977520*n^10-171862613511873758878288235394*n^9+ 192188202682655652758367119189*n^8-169115133125502420020916628271*n^7+ 115588544267200615669287330774*n^6-60217958197238781562156538724*n^5+ 23269099089634169136247659204*n^4-6404228497627538563660458096*n^3+ 1176760532756580208597973088*n^2-128287367020776725858129760*n+ 6227754420076623544320000)/n^2/(20526125081900992*n^15-777363231351272480*n^14+ 13583570035167327008*n^13-145257133420778334024*n^12+1062961284086716489884*n^ 11-5637785349407398551500*n^10+22386850193441871197155*n^9-\ 67764757106191690829351*n^8+157641633591197603653110*n^7-\ 281819608004724099992254*n^6+383998832909107706933795*n^5-\ 391625455736721914272295*n^4+289382480493137658782544*n^3-\ 146266116223527949810896*n^2+45220160739070155931272*n-6446909198345452637760)/ (3*n-1)^2/(3*n-2)^2*b(n-3)+1/36*(n-2)*(n-3)*(78738215814172205312*n^19-\ 3060703571277653438592*n^18+55002876150062012824512*n^17-\ 606533084140754111731072*n^16+4594173788016315630386800*n^15-\ 25354276222810749181953688*n^14+105535594162848712186915904*n^13-\ 338389379547709667253718636*n^12+846373970688528469585900607*n^11-\ 1662069261249598740203466314*n^10+2566740758243685350539302806*n^9-\ 3108733725291158422207230632*n^8+2932195647060171908233878263*n^7-\ 2128337289970643847072107750*n^6+1167666778279191106183552708*n^5-\ 471559350462241436713422660*n^4+134710377233222887428509568*n^3-\ 25533578219638220391747456*n^2+2855611262538225282172320*n-\ 141537579974663336755200)/n^2/(20526125081900992*n^15-777363231351272480*n^14+ 13583570035167327008*n^13-145257133420778334024*n^12+1062961284086716489884*n^ 11-5637785349407398551500*n^10+22386850193441871197155*n^9-\ 67764757106191690829351*n^8+157641633591197603653110*n^7-\ 281819608004724099992254*n^6+383998832909107706933795*n^5-\ 391625455736721914272295*n^4+289382480493137658782544*n^3-\ 146266116223527949810896*n^2+45220160739070155931272*n-6446909198345452637760)/ (3*n-1)^2/(3*n-2)^2*b(n-4)+11/12*(n-2)*(n-3)*(20526125081900992*n^15-\ 469471355122757600*n^14+4855727929849116448*n^13-30071390104303927240*n^12+ 124452090305359659580*n^11-363781641128706757576*n^10+774002613580364911215*n^9 -1217396791110327898100*n^8+1423097041295213928482*n^7-1232639346246718854812*n ^6+782185650712061928167*n^5-355959158795058424160*n^4+112101330751111620956*n^ 3-22993991052405656192*n^2+2739618485091811440*n-142809368673484800)*(n-4)^2*(2 *n-9)^2/n^2/(20526125081900992*n^15-777363231351272480*n^14+ 13583570035167327008*n^13-145257133420778334024*n^12+1062961284086716489884*n^ 11-5637785349407398551500*n^10+22386850193441871197155*n^9-\ 67764757106191690829351*n^8+157641633591197603653110*n^7-\ 281819608004724099992254*n^6+383998832909107706933795*n^5-\ 391625455736721914272295*n^4+289382480493137658782544*n^3-\ 146266116223527949810896*n^2+45220160739070155931272*n-6446909198345452637760)/ (3*n-1)^2/(3*n-2)^2*b(n-5) Of course, the initial conditions are b(0) = 1, b(1) = 33, b(2) = 3281, b(3) = 411885, b(4) = 57343617 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 21 20 a(n) = 1/18 (5462083988794181575168 n - 223245717281881556743424 n 19 18 + 4255004887063514446388544 n - 50258963098906439412928544 n 17 16 + 412380036293510751187713504 n - 2497092995372570094928567576 n 15 14 + 11572036430603475049748255208 n - 42002944238390782118831875114 n 13 12 + 121211009353311708102579577819 n - 280698794599642853983711044656 n 11 10 + 524276062550581932318282990315 n - 790933019434812438196740003020 n 9 8 + 962067661879164383038413522233 n - 938776069358884642417825306704 n 7 6 + 728521052293052544366044489945 n - 443792256052446436792915770122 n 5 4 + 208258243909832999896875112176 n - 73268588166745182041860584840 n 3 2 + 18558804977597965532024307888 n - 3172447797633475002711259200 n + 325170066792690929135289600 n - 14987450772753290625024000) a(n - 1) / 2 2 2 21 / (n %1 (3 n - 1) (3 n - 2) ) + 1/36 (4658855662089231956224 n / 20 19 - 195075109994617945155456 n + 3815248772647586074073280 n 18 17 - 46319126110040976546162240 n + 391286516119147423955760848 n 16 15 - 2443464565870966677944829816 n + 11696743043639881904438813464 n 14 13 - 43923920346742702717458485492 n + 131331441048677697873105114595 n 12 11 - 315536544154679255283235261567 n + 612132678688881432891158766910 n 10 9 - 960037432996574731806800688678 n + 1214700651816918316961542121935 n 8 7 - 1233190223476442593064249943139 n + 995431543843088742860488349288 n 6 5 - 630270921840367962724386035228 n + 307018090521789415948895980128 n 4 3 - 111912478526480431761073233072 n + 29298635452173767552952189408 n 2 - 5161009633717690120540830912 n + 543233520160201213208334720 n / 2 2 2 - 25614618339244074939187200) a(n - 2) / (n %1 (3 n - 1) (3 n - 2) ) / 20 19 + 1/36 (n - 2) (1568442269758218600704 n - 64105206043288088543872 n 18 17 + 1218695172261502724462272 n - 14312815398702836566154752 n 16 15 + 116322191053800465053775824 n - 694458971365097895127357464 n 14 13 + 3155452221421305861514977296 n - 11155676652688152456939335520 n 12 11 + 31109747724762666743867218529 n - 68969356669198846783434558947 n 10 9 + 121950513185622810631996977520 n - 171862613511873758878288235394 n 8 7 + 192188202682655652758367119189 n - 169115133125502420020916628271 n 6 5 + 115588544267200615669287330774 n - 60217958197238781562156538724 n 4 3 + 23269099089634169136247659204 n - 6404228497627538563660458096 n 2 + 1176760532756580208597973088 n - 128287367020776725858129760 n / 2 2 2 + 6227754420076623544320000) a(n - 3) / (n %1 (3 n - 1) (3 n - 2) ) + / 19 18 1/36 (n - 2) (n - 3) (78738215814172205312 n - 3060703571277653438592 n 17 16 + 55002876150062012824512 n - 606533084140754111731072 n 15 14 + 4594173788016315630386800 n - 25354276222810749181953688 n 13 12 + 105535594162848712186915904 n - 338389379547709667253718636 n 11 10 + 846373970688528469585900607 n - 1662069261249598740203466314 n 9 8 + 2566740758243685350539302806 n - 3108733725291158422207230632 n 7 6 + 2932195647060171908233878263 n - 2128337289970643847072107750 n 5 4 + 1167666778279191106183552708 n - 471559350462241436713422660 n 3 2 + 134710377233222887428509568 n - 25533578219638220391747456 n / 2 + 2855611262538225282172320 n - 141537579974663336755200) a(n - 4) / (n / 2 2 11 15 %1 (3 n - 1) (3 n - 2) ) + -- (n - 2) (n - 3) (20526125081900992 n 12 14 13 - 469471355122757600 n + 4855727929849116448 n 12 11 - 30071390104303927240 n + 124452090305359659580 n 10 9 - 363781641128706757576 n + 774002613580364911215 n 8 7 - 1217396791110327898100 n + 1423097041295213928482 n 6 5 - 1232639346246718854812 n + 782185650712061928167 n 4 3 - 355959158795058424160 n + 112101330751111620956 n 2 - 22993991052405656192 n + 2739618485091811440 n - 142809368673484800) 2 2 / 2 2 2 (n - 4) (2 n - 9) a(n - 5) / (n %1 (3 n - 1) (3 n - 2) ) / 15 14 13 %1 := 20526125081900992 n - 777363231351272480 n + 13583570035167327008 n 12 11 - 145257133420778334024 n + 1062961284086716489884 n 10 9 - 5637785349407398551500 n + 22386850193441871197155 n 8 7 - 67764757106191690829351 n + 157641633591197603653110 n 6 5 - 281819608004724099992254 n + 383998832909107706933795 n 4 3 - 391625455736721914272295 n + 289382480493137658782544 n 2 - 146266116223527949810896 n + 45220160739070155931272 n - 6446909198345452637760 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 0, a(4) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately -0.2722692737156423806743387390599656902774570742516490537499214617266654767\ -5439 518437335951366176449219715 10 This constant is identified as, 0 The implied delta is, -0.3467598971789061745739974859359786166757377272643853\ 963012719835945320830025649858092947100566592650 Since this is negative, there is no Apery-style irrationality proof of, 0, but still a very fast way to compute it to many digits ----------------------- This took, 3.512, seconds. ------------------------------------------------------------- The Apery limit generated by the binomial coefficient sum, ----- \ 2 k ) binomial(n, k) binomial(2 n + 2 k, k) 3 / ----- k By Shalosh B. Ekhad Let ----- \ 2 k b(n) = ) binomial(n, k) binomial(2 n + 2 k, k) 3 / ----- k The famous Zeilberger algorithm finds (and proves!, but proof omitted) that , b(n), sastisfies the following linear recurrence with polynomial coefficients 21 20 b(n) = 2/27 (600539081452648833574944 n - 24545767659461926227087840 n 19 18 + 467851679715583242708838344 n - 5526374662955836042682477400 n 17 16 + 45346847319222325821793507728 n - 274606521168271086340353203742 n 15 + 1272672435851365517534734518174 n 14 - 4619781922876846674972970702364 n 13 + 13332853569219700786998931896831 n 12 - 30879171837614938658433451742738 n 11 + 57681220274615750475562571714799 n 10 - 87030079608072921980671275920018 n 9 + 105875973410665600120879271786685 n 8 - 103329411821285249998542670808608 n 7 + 80201713552707808578120853209687 n 6 - 48866581506662064007233311866786 n 5 4 + 22937073280848097776125560273080 n - 8071821508538026576657708352488 n 3 2 + 2045202918114335115772599605808 n - 349724081362480092330417205056 n + 35858145585352893671641248000 n - 1653196413604138101419212800) b(n - 1) / 2 2 2 21 / (n %1 (3 n - 1) (3 n - 2) ) + 2/729 (4875508610449900205981760 n / 20 19 - 204151634461129065726735360 n + 3992815901294072868682287024 n 18 17 - 48474804422396179722740738424 n + 409489267821347702055293437440 n 16 - 2557037783729002072419062345956 n 15 + 12239679773151905524736095949908 n 14 - 45958939597308318256529667195100 n 13 + 137401271253808800007125183363607 n 12 - 330076133468003104774822898542600 n 11 + 640239056338091069044348928709145 n 10 - 1003939839649072678216003094442230 n 9 + 1270001671015780975562330626886297 n 8 - 1289068051193192933124924238553432 n 7 + 1040319479730849992233192176048683 n 6 - 658560150012336156278955606533738 n 5 + 320740226881192571919108272524376 n 4 - 116896966412128241677041649701192 n 3 2 + 30600415331745157568935325170800 n - 5390119123112717966625076029888 n + 567389041373460308707704061440 n - 26760130639339121791563202560) / 2 2 2 b(n - 2) / (n %1 (3 n - 1) (3 n - 2) ) + 4/729 (n - 2) ( / 20 19 456144198124570376942592 n - 18643931514488928201477120 n 18 17 + 354458694060894394485503856 n - 4163305472624175535715256864 n 16 15 + 33840388449703942314644762160 n - 202067454480317657483136982896 n 14 13 + 918346075531045952475314505396 n - 3247536939928240322177269186459 n 12 + 9059137911200239700983716445491 n 11 - 20090825264233821893289153760922 n 10 + 35538212551595249965937097878064 n 9 - 50105202212025733904552066537415 n 8 + 56057870468951254740593223679947 n 7 - 49353694167001150464703179357836 n 6 + 33751890230074099233073682310510 n 5 4 - 17594399393510939901841332277000 n + 6803197019725179081132956563536 n 3 2 - 1873733067488299393596388922208 n + 344558383979355499557223689408 n - 37593862254337016958079985280 n + 1826365966106445947040399360) b(n - 3) / 2 2 2 / (n %1 (3 n - 1) (3 n - 2) ) + 4/729 (n - 2) (n - 3) ( / 19 18 17796974651170684766208 n - 691819833281620205758464 n 17 16 + 12432661227658711604980512 n - 137100110032808484018496272 n 15 14 + 1038472262270837502195963696 n - 5731185075561819278545981836 n 13 12 + 23856213775021477235754310068 n - 76495222292713191704397039412 n 11 10 + 191338651176346223103763871131 n - 375770172859355270083373281038 n 9 8 + 580363639065547880163906332238 n - 703009838608017688622828289192 n 7 6 + 663205285809791545793565073647 n - 481496721021827522633001602738 n 5 4 + 264237329327061298959764610356 n - 106748846277081195105356743656 n 3 2 + 30508063411665139056710109072 n - 5785677784542304190182606272 n + 647460210174825310421844672 n - 32109646296903573209333760) b(n - 4) / 2 2 2 196 / (n %1 (3 n - 1) (3 n - 2) ) + --- (n - 2) (n - 3) ( / 729 15 14 2015056006699579344 n - 46090153856758293648 n 13 12 + 476743542723868046064 n - 2952761848030212260940 n 11 10 + 12221795846939939383236 n - 35731025646254895280084 n 9 8 + 76038525308460127325619 n - 119626064078284855312452 n 7 6 + 139877638226614341719334 n - 121196584037053989665724 n 5 4 + 76935899947216448339271 n - 35027854303650867041168 n 3 2 + 11037086618212273316268 n - 2265338665443596998848 n 2 2 + 270102997068444530928 n - 14088808724441794560) (n - 4) (2 n - 9) / 2 15 14 b(n - 5) / (n (2015056006699579344 n - 76315993957251983808 n / 13 12 + 1333586577421939988256 n - 14261482387453810183260 n 11 10 + 104368301808769138799940 n - 553589172350348387603704 n 9 8 + 2198384636301679595047015 n - 6655026932376516480433995 n 7 6 + 15483016365391817132608074 n - 27682028992965211276270434 n 5 4 + 37722627752655750335203311 n - 38476135685358949533932495 n 3 2 + 28434421554840423289128644 n - 14373767581272692787200064 n 2 + 4444437476349078714346104 n - 633719615084541758887488) (3 n - 1) 2 (3 n - 2) ) 15 14 %1 := 2015056006699579344 n - 76315993957251983808 n 13 12 + 1333586577421939988256 n - 14261482387453810183260 n 11 10 + 104368301808769138799940 n - 553589172350348387603704 n 9 8 + 2198384636301679595047015 n - 6655026932376516480433995 n 7 6 + 15483016365391817132608074 n - 27682028992965211276270434 n 5 4 + 37722627752655750335203311 n - 38476135685358949533932495 n 3 2 + 28434421554840423289128644 n - 14373767581272692787200064 n + 4444437476349078714346104 n - 633719615084541758887488 and in Maple notation b(n) = 2/27*(600539081452648833574944*n^21-24545767659461926227087840*n^20+ 467851679715583242708838344*n^19-5526374662955836042682477400*n^18+ 45346847319222325821793507728*n^17-274606521168271086340353203742*n^16+ 1272672435851365517534734518174*n^15-4619781922876846674972970702364*n^14+ 13332853569219700786998931896831*n^13-30879171837614938658433451742738*n^12+ 57681220274615750475562571714799*n^11-87030079608072921980671275920018*n^10+ 105875973410665600120879271786685*n^9-103329411821285249998542670808608*n^8+ 80201713552707808578120853209687*n^7-48866581506662064007233311866786*n^6+ 22937073280848097776125560273080*n^5-8071821508538026576657708352488*n^4+ 2045202918114335115772599605808*n^3-349724081362480092330417205056*n^2+ 35858145585352893671641248000*n-1653196413604138101419212800)/n^2/( 2015056006699579344*n^15-76315993957251983808*n^14+1333586577421939988256*n^13-\ 14261482387453810183260*n^12+104368301808769138799940*n^11-\ 553589172350348387603704*n^10+2198384636301679595047015*n^9-\ 6655026932376516480433995*n^8+15483016365391817132608074*n^7-\ 27682028992965211276270434*n^6+37722627752655750335203311*n^5-\ 38476135685358949533932495*n^4+28434421554840423289128644*n^3-\ 14373767581272692787200064*n^2+4444437476349078714346104*n-\ 633719615084541758887488)/(3*n-1)^2/(3*n-2)^2*b(n-1)+2/729*( 4875508610449900205981760*n^21-204151634461129065726735360*n^20+ 3992815901294072868682287024*n^19-48474804422396179722740738424*n^18+ 409489267821347702055293437440*n^17-2557037783729002072419062345956*n^16+ 12239679773151905524736095949908*n^15-45958939597308318256529667195100*n^14+ 137401271253808800007125183363607*n^13-330076133468003104774822898542600*n^12+ 640239056338091069044348928709145*n^11-1003939839649072678216003094442230*n^10+ 1270001671015780975562330626886297*n^9-1289068051193192933124924238553432*n^8+ 1040319479730849992233192176048683*n^7-658560150012336156278955606533738*n^6+ 320740226881192571919108272524376*n^5-116896966412128241677041649701192*n^4+ 30600415331745157568935325170800*n^3-5390119123112717966625076029888*n^2+ 567389041373460308707704061440*n-26760130639339121791563202560)/n^2/( 2015056006699579344*n^15-76315993957251983808*n^14+1333586577421939988256*n^13-\ 14261482387453810183260*n^12+104368301808769138799940*n^11-\ 553589172350348387603704*n^10+2198384636301679595047015*n^9-\ 6655026932376516480433995*n^8+15483016365391817132608074*n^7-\ 27682028992965211276270434*n^6+37722627752655750335203311*n^5-\ 38476135685358949533932495*n^4+28434421554840423289128644*n^3-\ 14373767581272692787200064*n^2+4444437476349078714346104*n-\ 633719615084541758887488)/(3*n-1)^2/(3*n-2)^2*b(n-2)+4/729*(n-2)*( 456144198124570376942592*n^20-18643931514488928201477120*n^19+ 354458694060894394485503856*n^18-4163305472624175535715256864*n^17+ 33840388449703942314644762160*n^16-202067454480317657483136982896*n^15+ 918346075531045952475314505396*n^14-3247536939928240322177269186459*n^13+ 9059137911200239700983716445491*n^12-20090825264233821893289153760922*n^11+ 35538212551595249965937097878064*n^10-50105202212025733904552066537415*n^9+ 56057870468951254740593223679947*n^8-49353694167001150464703179357836*n^7+ 33751890230074099233073682310510*n^6-17594399393510939901841332277000*n^5+ 6803197019725179081132956563536*n^4-1873733067488299393596388922208*n^3+ 344558383979355499557223689408*n^2-37593862254337016958079985280*n+ 1826365966106445947040399360)/n^2/(2015056006699579344*n^15-\ 76315993957251983808*n^14+1333586577421939988256*n^13-14261482387453810183260*n ^12+104368301808769138799940*n^11-553589172350348387603704*n^10+ 2198384636301679595047015*n^9-6655026932376516480433995*n^8+ 15483016365391817132608074*n^7-27682028992965211276270434*n^6+ 37722627752655750335203311*n^5-38476135685358949533932495*n^4+ 28434421554840423289128644*n^3-14373767581272692787200064*n^2+ 4444437476349078714346104*n-633719615084541758887488)/(3*n-1)^2/(3*n-2)^2*b(n-3 )+4/729*(n-2)*(n-3)*(17796974651170684766208*n^19-691819833281620205758464*n^18 +12432661227658711604980512*n^17-137100110032808484018496272*n^16+ 1038472262270837502195963696*n^15-5731185075561819278545981836*n^14+ 23856213775021477235754310068*n^13-76495222292713191704397039412*n^12+ 191338651176346223103763871131*n^11-375770172859355270083373281038*n^10+ 580363639065547880163906332238*n^9-703009838608017688622828289192*n^8+ 663205285809791545793565073647*n^7-481496721021827522633001602738*n^6+ 264237329327061298959764610356*n^5-106748846277081195105356743656*n^4+ 30508063411665139056710109072*n^3-5785677784542304190182606272*n^2+ 647460210174825310421844672*n-32109646296903573209333760)/n^2/( 2015056006699579344*n^15-76315993957251983808*n^14+1333586577421939988256*n^13-\ 14261482387453810183260*n^12+104368301808769138799940*n^11-\ 553589172350348387603704*n^10+2198384636301679595047015*n^9-\ 6655026932376516480433995*n^8+15483016365391817132608074*n^7-\ 27682028992965211276270434*n^6+37722627752655750335203311*n^5-\ 38476135685358949533932495*n^4+28434421554840423289128644*n^3-\ 14373767581272692787200064*n^2+4444437476349078714346104*n-\ 633719615084541758887488)/(3*n-1)^2/(3*n-2)^2*b(n-4)+196/729*(n-2)*(n-3)*( 2015056006699579344*n^15-46090153856758293648*n^14+476743542723868046064*n^13-\ 2952761848030212260940*n^12+12221795846939939383236*n^11-\ 35731025646254895280084*n^10+76038525308460127325619*n^9-\ 119626064078284855312452*n^8+139877638226614341719334*n^7-\ 121196584037053989665724*n^6+76935899947216448339271*n^5-\ 35027854303650867041168*n^4+11037086618212273316268*n^3-2265338665443596998848* n^2+270102997068444530928*n-14088808724441794560)*(n-4)^2*(2*n-9)^2/n^2/( 2015056006699579344*n^15-76315993957251983808*n^14+1333586577421939988256*n^13-\ 14261482387453810183260*n^12+104368301808769138799940*n^11-\ 553589172350348387603704*n^10+2198384636301679595047015*n^9-\ 6655026932376516480433995*n^8+15483016365391817132608074*n^7-\ 27682028992965211276270434*n^6+37722627752655750335203311*n^5-\ 38476135685358949533932495*n^4+28434421554840423289128644*n^3-\ 14373767581272692787200064*n^2+4444437476349078714346104*n-\ 633719615084541758887488)/(3*n-1)^2/(3*n-2)^2*b(n-5) Of course, the initial conditions are b(0) = 1, b(1) = 49, b(2) = 7273, b(3) = 1362052, b(4) = 282850393 Let's consider the related sequence, let's call it, a(n) satisying the same recurrence, i.e. 21 20 a(n) = 2/27 (600539081452648833574944 n - 24545767659461926227087840 n 19 18 + 467851679715583242708838344 n - 5526374662955836042682477400 n 17 16 + 45346847319222325821793507728 n - 274606521168271086340353203742 n 15 + 1272672435851365517534734518174 n 14 - 4619781922876846674972970702364 n 13 + 13332853569219700786998931896831 n 12 - 30879171837614938658433451742738 n 11 + 57681220274615750475562571714799 n 10 - 87030079608072921980671275920018 n 9 + 105875973410665600120879271786685 n 8 - 103329411821285249998542670808608 n 7 + 80201713552707808578120853209687 n 6 - 48866581506662064007233311866786 n 5 4 + 22937073280848097776125560273080 n - 8071821508538026576657708352488 n 3 2 + 2045202918114335115772599605808 n - 349724081362480092330417205056 n + 35858145585352893671641248000 n - 1653196413604138101419212800) a(n - 1) / 2 2 2 21 / (n %1 (3 n - 1) (3 n - 2) ) + 2/729 (4875508610449900205981760 n / 20 19 - 204151634461129065726735360 n + 3992815901294072868682287024 n 18 17 - 48474804422396179722740738424 n + 409489267821347702055293437440 n 16 - 2557037783729002072419062345956 n 15 + 12239679773151905524736095949908 n 14 - 45958939597308318256529667195100 n 13 + 137401271253808800007125183363607 n 12 - 330076133468003104774822898542600 n 11 + 640239056338091069044348928709145 n 10 - 1003939839649072678216003094442230 n 9 + 1270001671015780975562330626886297 n 8 - 1289068051193192933124924238553432 n 7 + 1040319479730849992233192176048683 n 6 - 658560150012336156278955606533738 n 5 + 320740226881192571919108272524376 n 4 - 116896966412128241677041649701192 n 3 2 + 30600415331745157568935325170800 n - 5390119123112717966625076029888 n + 567389041373460308707704061440 n - 26760130639339121791563202560) / 2 2 2 a(n - 2) / (n %1 (3 n - 1) (3 n - 2) ) + 4/729 (n - 2) ( / 20 19 456144198124570376942592 n - 18643931514488928201477120 n 18 17 + 354458694060894394485503856 n - 4163305472624175535715256864 n 16 15 + 33840388449703942314644762160 n - 202067454480317657483136982896 n 14 13 + 918346075531045952475314505396 n - 3247536939928240322177269186459 n 12 + 9059137911200239700983716445491 n 11 - 20090825264233821893289153760922 n 10 + 35538212551595249965937097878064 n 9 - 50105202212025733904552066537415 n 8 + 56057870468951254740593223679947 n 7 - 49353694167001150464703179357836 n 6 + 33751890230074099233073682310510 n 5 4 - 17594399393510939901841332277000 n + 6803197019725179081132956563536 n 3 2 - 1873733067488299393596388922208 n + 344558383979355499557223689408 n - 37593862254337016958079985280 n + 1826365966106445947040399360) a(n - 3) / 2 2 2 / (n %1 (3 n - 1) (3 n - 2) ) + 4/729 (n - 2) (n - 3) ( / 19 18 17796974651170684766208 n - 691819833281620205758464 n 17 16 + 12432661227658711604980512 n - 137100110032808484018496272 n 15 14 + 1038472262270837502195963696 n - 5731185075561819278545981836 n 13 12 + 23856213775021477235754310068 n - 76495222292713191704397039412 n 11 10 + 191338651176346223103763871131 n - 375770172859355270083373281038 n 9 8 + 580363639065547880163906332238 n - 703009838608017688622828289192 n 7 6 + 663205285809791545793565073647 n - 481496721021827522633001602738 n 5 4 + 264237329327061298959764610356 n - 106748846277081195105356743656 n 3 2 + 30508063411665139056710109072 n - 5785677784542304190182606272 n + 647460210174825310421844672 n - 32109646296903573209333760) a(n - 4) / 2 2 2 196 / (n %1 (3 n - 1) (3 n - 2) ) + --- (n - 2) (n - 3) ( / 729 15 14 2015056006699579344 n - 46090153856758293648 n 13 12 + 476743542723868046064 n - 2952761848030212260940 n 11 10 + 12221795846939939383236 n - 35731025646254895280084 n 9 8 + 76038525308460127325619 n - 119626064078284855312452 n 7 6 + 139877638226614341719334 n - 121196584037053989665724 n 5 4 + 76935899947216448339271 n - 35027854303650867041168 n 3 2 + 11037086618212273316268 n - 2265338665443596998848 n 2 2 + 270102997068444530928 n - 14088808724441794560) (n - 4) (2 n - 9) / 2 15 14 a(n - 5) / (n (2015056006699579344 n - 76315993957251983808 n / 13 12 + 1333586577421939988256 n - 14261482387453810183260 n 11 10 + 104368301808769138799940 n - 553589172350348387603704 n 9 8 + 2198384636301679595047015 n - 6655026932376516480433995 n 7 6 + 15483016365391817132608074 n - 27682028992965211276270434 n 5 4 + 37722627752655750335203311 n - 38476135685358949533932495 n 3 2 + 28434421554840423289128644 n - 14373767581272692787200064 n 2 + 4444437476349078714346104 n - 633719615084541758887488) (3 n - 1) 2 (3 n - 2) ) 15 14 %1 := 2015056006699579344 n - 76315993957251983808 n 13 12 + 1333586577421939988256 n - 14261482387453810183260 n 11 10 + 104368301808769138799940 n - 553589172350348387603704 n 9 8 + 2198384636301679595047015 n - 6655026932376516480433995 n 7 6 + 15483016365391817132608074 n - 27682028992965211276270434 n 5 4 + 37722627752655750335203311 n - 38476135685358949533932495 n 3 2 + 28434421554840423289128644 n - 14373767581272692787200064 n + 4444437476349078714346104 n - 633719615084541758887488 but with the following simpler initial conditions a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 0, a(4) = 1 Using the recurrence, we can compute many terms, how about, 2000, terms for e\ ach sequence and get a very good approximation to the Apery limit, i.e. \ a(n) the limit of, ----, as n goes to infinity b(n) that is approximately 0.53953504328915115998826177040079121974232618060680684342641211180824666544\ -6227 21466199585929184158285858 10 This constant is identified as, 0 The implied delta is, -0.3359274660250243711344285207831352093264379887770413\ 203068216533926062224834584614072307541328827801 Since this is negative, there is no Apery-style irrationality proof of, 0, but still a very fast way to compute it to many digits ----------------------- This took, 3.485, seconds. ---------------------------------- This ends this book that took altogether, 295.800, seconds to produce.