This is a collection of candidates for Apery-style proofs using generalized Alladi-Robinson integrals Each entry is a list of length 7 where The first entry is the definition of the constant as a 2F1 The second entry is its value to 100 decimals The third entry is the second-order linear recurrence operator in n and N a\ nnihilating Both sequence of rational numbers A(n) and B(n) such that that A(n)/B(n) converges to the constant The fourth entry is the pair [[A(0),A[1]],[B(0),B(1)]], of initial condition\ s enabling the fast computations of A(n) and B(n) using the recurrence The fifth entry is the estimate for the limit of log(E(n))/n as n goes to in\ finity of the INTEGERating factor, i.e. the sequence of rational numbers such that A(n)*E(n) and B(n)*E(n) are BOTH integers The sixth entry is the estimate for the implied irrationality measure. The seventh entry is the triple [a,b,c] such that the constant is Int(x^a*(1\ -x)^b/(1+c*x),x=0..1)/Beta(a+1,b+1). If Maple can identify the constant, then there is a comment to that effect. denominator , 1 [`2F1(1,1;2;-1)`, .693147180559945309417232121458176568075500134360255254120680\ 0094933936219696947156058633269964186875, (1+n)/(n+2)-3*(2*n+3)/(n+2)*N+N^2, [[ 0, 2], [1, 3]], .9878596631, 4.549685358, [0, 0, 1]] Comment: This probably equals, arcsinh(3/4) ----------------------- denominator , 2 [`2F1(1,1;2;-1/2)`, .8109302162163287639560262309286982731439808469249883952280\ 286482882013424978285025355048556346268025, 4*(1+n)/(n+2)-10*(2*n+3)/(n+2)*N+N^ 2, [[0, 8], [1, 10]], .3046096875, 3.541324295, [0, 0, 1/2]] 65 Comment: This probably equals, arcsinh(--) 72 [`2F1(1,3/2;3;-1/2)`, .80816411546915042884345480470573772854484030949327794507\ 69189239846339606829595753622490703257562429, (5+2*n)*(2*n+3)/(3+n)/(n+2)-10*(5 +2*n)/(3+n)*N+N^2, [[0, 12], [1, 15]], -.6918529773, 2.001129746, [1/2, 1/2, 1/ 2]] 1/2 Comment: This probably equals, 40 - 16 6 [`2F1(1,3/2;3;-1)`, .6862915010152396095864902063224153714426249969844154145865\ 620960741401723031436891968997253788674181, 1/4*(5+2*n)*(2*n+3)/(3+n)/(n+2)-3*( 5+2*n)/(3+n)*N+N^2, [[0, 3], [1, 9/2]], .1294203279e-2, 2.001469474, [1/2, 1/2, 1]] 1/2 Comment: This probably equals, 12 - 8 2 ----------------------- denominator , 3 [`2F1(1,1;2;-1/3)`, .8630462173553427823176570179814822945105291326932831695199\ 970560478788521623413930143326975373158589, 9*(1+n)/(n+2)-21*(2*n+3)/(n+2)*N+N^ 2, [[0, 18], [1, 21]], -.1059997156, 3.209541921, [0, 0, 1/3]] 3367 Comment: This probably equals, arcsinh(----) 3456 [`2F1(1,1;2;-2/3)`, .7662384356489860248082711444554929023171661946686524052669\ 303367550270417335731966347771849191712830, 9/4*(1+n)/(n+2)-6*(2*n+3)/(n+2)*N+N ^2, [[0, 9/2], [1, 6]], 1.281028017, 10.94824370, [0, 0, 2/3]] 49 Comment: This probably equals, arctanh(--) 76 [`2F1(1,4/3;8/3;-1/3)`, .861965837554122126446039279709070855387994024432155132\ 2465315926076327230222382934477146584570572557, 3*(3*n+7)*(3*n+4)/(8+3*n)/(n+2) -7*(3*n+7)*(6*n+11)/(8+3*n)/(n+2)*N+N^2, [[0, 24], [1, 28]], .7786683170, 6.962\ 181267, [1/3, 1/3, 1/3]] [`2F1(1,4/3;8/3;-1)`, .68815073461701577507747495811464727057023580859933070607\ 45774307898875090732592492655928535256332886, 1/3*(3*n+7)*(3*n+4)/(8+3*n)/(n+2) -(3*n+7)*(6*n+11)/(8+3*n)/(n+2)*N+N^2, [[0, 8/3], [1, 4]], 1.056328987, 4.99066\ 1947, [1/3, 1/3, 1]] [`2F1(1,4/3;3;-1/3)`, .87548798706445841987456515832793286887663499037413774000\ 79479117092740691976992550543479771734781052, (3*n+5)*(5+2*n)*(3*n+4)/(n+2)/(2* n+3)/(3+n)-2*(42*n^2+168*n+157)/(3+n)/(2*n+3)*N+N^2, [[0, 27], [1, 31]], -.2005\ 232240, 3.034768052, [1/3, 2/3, 1/3]] (1/3) Comment: This probably equals, -54 6 + 99 [`2F1(1,4/3;3;-2/3)`, .78414754606596305395729589004503816564606571155645750056\ 33960208576676266778611282284542389253830052, 1/4*(3*n+5)*(5+2*n)*(3*n+4)/(n+2) /(2*n+3)/(3+n)-1/2*(48*n^2+192*n+179)/(3+n)/(2*n+3)*N+N^2, [[0, 27/4], [1, 35/4 ]], 1.185649782, 8.737413438, [1/3, 2/3, 2/3]] 1/3 27 75 Comment: This probably equals, - -------- + 117/4 4 [`2F1(1,4/3;3;-1)`, .7133905322862047272346492465492256564765600489013229117254\ 281435730514493820274405381020806400113868, 1/9*(3*n+5)*(5+2*n)*(3*n+4)/(n+2)/( 2*n+3)/(3+n)-2/3*(18*n^2+72*n+67)/(3+n)/(2*n+3)*N+N^2, [[0, 3], [1, 13/3]], .90\ 18948995, 4.095353483, [1/3, 2/3, 1]] (2/3) Comment: This probably equals, -9 2 + 15 [`2F1(1,5/3;3;-1/3)`, .84796427984507953538553128495470616513411460340067990172\ 35314980495567593704685316381772305501448099, (3*n+5)*(5+2*n)*(3*n+4)/(n+2)/(2* n+3)/(3+n)-4*(21*n^2+84*n+79)/(3+n)/(2*n+3)*N+N^2, [[0, 27], [1, 32]], -.196717\ 3892, 3.041436462, [2/3, 1/3, 1/3]] (2/3) Comment: This probably equals, -27 6 + 90 [`2F1(1,5/3;3;-2/3)`, .74097019469207605945845629988734680139791332712880277270\ 03993816757249706029565240046506437844589692, 1/4*(3*n+5)*(5+2*n)*(3*n+4)/(n+2) /(2*n+3)/(3+n)-1/2*(48*n^2+192*n+181)/(3+n)/(2*n+3)*N+N^2, [[0, 27/4], [1, 37/4 ]], 1.189576972, 8.810671015, [2/3, 1/3, 2/3]] 1/3 27 45 Comment: This probably equals, - -------- + 99/4 4 [`2F1(1,5/3;3;-1)`, .6607105509461415170951045344959448448677368176864281792622\ 239906023029113743646464354309380737041526, 1/9*(3*n+5)*(5+2*n)*(3*n+4)/(n+2)/( 2*n+3)/(3+n)-4/3*(9*n^2+36*n+34)/(3+n)/(2*n+3)*N+N^2, [[0, 3], [1, 14/3]], .901\ 8948995, 4.095353483, [2/3, 1/3, 1]] (1/3) Comment: This probably equals, -9 2 + 12 [`2F1(1,5/3;10/3;-1/3)`, .86121914488406966155483521453076060189883018335907679\ 19435509833777417581072717833470066022052656129, 3*(3*n+8)*(3*n+5)/(10+3*n)/(n+ 2)-7*(3*n+8)*(6*n+13)/(10+3*n)/(n+2)*N+N^2, [[0, 30], [1, 35]], .7569999026, 6.\ 768350196, [2/3, 2/3, 1/3]] [`2F1(1,5/3;10/3;-1)`, .6847252627218201717748664466479410517588445620320973267\ 260420904660066381273769275040341346048265071, 1/3*(3*n+8)*(3*n+5)/(10+3*n)/(n+ 2)-(3*n+8)*(6*n+13)/(10+3*n)/(n+2)*N+N^2, [[0, 10/3], [1, 5]], 1.033467114, 4.8\ 34211926, [2/3, 2/3, 1]] ----------------------- denominator , 4 [`2F1(1,1;2;-1/4)`, .8925742052568390230651803612393380134984043421920288546851\ 514899495669750730733336736288964013689429, 16*(1+n)/(n+2)-36*(2*n+3)/(n+2)*N+N ^2, [[0, 32], [1, 36]], -.3921703656, 3.050234413, [0, 0, 1/4]] 325089 Comment: This probably equals, arcsinh(------) 320000 [`2F1(1,1;5/2;-1/4)`, .91226822892012116668299698851816276273712047620868964271\ 01831197768364638339588408447079409165986622, 16*(2*n+3)*(4*n+9)*(1+n)/(n+2)/(4 *n+5)/(5+2*n)-4*(4*n+7)*(36*n^2+126*n+101)/(n+2)/(4*n+5)/(5+2*n)*N+N^2, [[0, 40 ], [1, 44]], .5938695114, 6.365888091, [0, 1/2, 1/4]] [`2F1(1,5/4;5/2;-1/4)`, .892045588680414870893981729966919693583128266037554663\ 4052341222440079035379247571038583157219871290, 2*(4*n+9)*(4*n+5)/(5+2*n)/(n+2) -9*(4*n+9)*(4*n+7)/(5+2*n)/(n+2)*N+N^2, [[0, 40], [1, 45]], .3066011546, 4.8347\ 79674, [1/4, 1/4, 1/4]] [`2F1(1,5/4;5/2;-1/2)`, .809347819001111648408878445725322725430017224859767879\ 1851192900059769587999132899622967035230452200, 1/2*(4*n+9)*(4*n+5)/(5+2*n)/(n+ 2)-5/2*(4*n+9)*(4*n+7)/(5+2*n)/(n+2)*N+N^2, [[0, 10], [1, 25/2]], 1.000025118, 7.650882964, [1/4, 1/4, 1/2]] [`2F1(1,5/4;5/2;-1)`, .68921689098230447418762512352920885201207164728671763964\ 33415444367177652514540546123050911060433595, 1/8*(4*n+9)*(4*n+5)/(5+2*n)/(n+2) -3/4*(4*n+9)*(4*n+7)/(5+2*n)/(n+2)*N+N^2, [[0, 5/2], [1, 15/4]], 1.688691743, 47.60615781, [1/4, 1/4, 1]] [`2F1(1,5/4;3;-1/4)`, .90845674598915982897149442406079561169134877616207756545\ 97669839594121531594477420667899594455039679, (4*n+7)*(5+2*n)*(4*n+5)/(n+2)/(2* n+3)/(3+n)-2*(72*n^2+288*n+269)/(3+n)/(2*n+3)*N+N^2, [[0, 48], [1, 53]], .17888\ 17060, 4.367721354, [1/4, 3/4, 1/4]] [`2F1(1,5/4;3;-1/2)`, .83613843563659742108471128161423849245757606806548166158\ 99976802067328624248676374760397614676567460, 1/4*(4*n+7)*(5+2*n)*(4*n+5)/(n+2) /(2*n+3)/(3+n)-(40*n^2+160*n+149)/(3+n)/(2*n+3)*N+N^2, [[0, 12], [1, 29/2]], .8\ 700168862, 6.286942270, [1/4, 3/4, 1/2]] 1/4 3/4 64 2 3 Comment: This probably equals, - ------------ + 176/3 3 [`2F1(1,5/4;3;-1)`, .7275431412540897486693231736914155724792690697219304359845\ 101658682718943167522172944212437484258950, 1/16*(4*n+7)*(5+2*n)*(4*n+5)/(n+2)/ (2*n+3)/(3+n)-1/2*(24*n^2+96*n+89)/(3+n)/(2*n+3)*N+N^2, [[0, 3], [1, 17/4]], 1.\ 564725020, 17.80353533, [1/4, 3/4, 1]] 3/4 32 2 Comment: This probably equals, - ------- + 56/3 3 [`2F1(1,3/2;11/4;-1/4)`, .88285812206300826130452632739408010309377510550831749\ 84334254628633944779463022894758526132144921862, 8*(4*n+5)*(2*n+3)*(8*n+19)/(n+ 2)/(8*n+11)/(11+4*n)-6*(8*n+15)*(48*n^2+180*n+157)/(n+2)/(8*n+11)/(11+4*n)*N+N^ 2, [[0, 44], [1, 50]], 1.295074635, 28.04511081, [1/2, 1/4, 1/4]] [`2F1(1,3/2;3;-1/4)`, .89164944001345942981288520119832093180042498486235364666\ 25762937266607591804864374774778637579344020, 4*(5+2*n)*(2*n+3)/(3+n)/(n+2)-36* (5+2*n)/(3+n)*N+N^2, [[0, 48], [1, 54]], -1.385000158, 2.000896906, [1/2, 1/2, 1/4]] 1/2 Comment: This probably equals, 144 - 64 5 [`2F1(1,3/2;3;-3/4)`, .74132401020735580087739908523192586161593469808035427293\ 62882901257328125846497581483172517365787690, 4/9*(5+2*n)*(2*n+3)/(3+n)/(n+2)-\ 44/9*(5+2*n)/(3+n)*N+N^2, [[0, 16/3], [1, 22/3]], .8122244195, 4.518713824, [1/ 2, 1/2, 3/4]] 1/2 64 7 Comment: This probably equals, 176/9 - ------- 9 [`2F1(1,3/2;13/4;-1/4)`, .89918398368925455454113886808089778948684111229040269\ 10237998746201390003384291723474607698077747102, 8*(2*n+3)*(4*n+7)*(8*n+21)/(n+ 2)/(8*n+13)/(13+4*n)-2*(8*n+17)*(144*n^2+612*n+613)/(n+2)/(8*n+13)/(13+4*n)*N+N ^2, [[0, 52], [1, 58]], 1.301248065, 28.91196029, [1/2, 3/4, 1/4]] [`2F1(1,7/4;3;-1/4)`, .87530437281039026602035199511795592856252411813497016122\ 22200806065783735447741130776077043855258596, (4*n+7)*(5+2*n)*(4*n+5)/(n+2)/(2* n+3)/(3+n)-2*(72*n^2+288*n+271)/(3+n)/(2*n+3)*N+N^2, [[0, 48], [1, 55]], .17992\ 68708, 4.371176931, [3/4, 1/4, 1/4]] [`2F1(1,7/4;3;-1/2)`, .78157142611961205722494833321251632836615754539052904461\ 56712671983579585245938455892872761730101823, 1/4*(4*n+7)*(5+2*n)*(4*n+5)/(n+2) /(2*n+3)/(3+n)-(40*n^2+160*n+151)/(3+n)/(2*n+3)*N+N^2, [[0, 12], [1, 31/2]], .8\ 706863272, 6.292718754, [3/4, 1/4, 1/2]] 3/4 1/4 64 2 3 Comment: This probably equals, - ------------ + 48 3 [`2F1(1,7/4;3;-1)`, .6484574399709752883466669806882569035416310137192809277973\ 096029923555389128836280449989915931981470, 1/16*(4*n+7)*(5+2*n)*(4*n+5)/(n+2)/ (2*n+3)/(3+n)-1/2*(24*n^2+96*n+91)/(3+n)/(2*n+3)*N+N^2, [[0, 3], [1, 19/4]], 1.\ 558537404, 17.26408297, [3/4, 1/4, 1]] 1/4 32 2 Comment: This probably equals, - ------- + 40/3 3 [`2F1(1,7/4;7/2;-1/4)`, .891341538739049417433195126148382587843270524857867632\ 2251129653719658230292238240537874055774571156, 2*(4*n+11)*(4*n+7)/(7+2*n)/(n+2 )-9*(4*n+11)*(4*n+9)/(7+2*n)/(n+2)*N+N^2, [[0, 56], [1, 63]], .2901373283, 4.76\ 9040991, [3/4, 3/4, 1/4]] [`2F1(1,7/4;7/2;-1/2)`, .807245566971563966191108563636390524259955772508134140\ 9528433631314906223016099943509133203255102072, 1/2*(4*n+11)*(4*n+7)/(7+2*n)/(n +2)-5/2*(4*n+11)*(4*n+9)/(7+2*n)/(n+2)*N+N^2, [[0, 14], [1, 35/2]], .9789890832 , 7.391419104, [3/4, 3/4, 1/2]] [`2F1(1,7/4;7/2;-1)`, .68403133820765647019899085122451846910758922195652025039\ 71633818535843029480955486999406500067396696, 1/8*(4*n+11)*(4*n+7)/(7+2*n)/(n+2 )-3/4*(4*n+11)*(4*n+9)/(7+2*n)/(n+2)*N+N^2, [[0, 7/2], [1, 21/4]], 1.678434124, 41.81433967, [3/4, 3/4, 1]] ----------------------- denominator , 5 [`2F1(1,1;2;-1/5)`, .9116077839697731310585901257725731659869466895724349197136\ 322582835446374032295892467260185848558276, 25*(1+n)/(n+2)-55*(2*n+3)/(n+2)*N+N ^2, [[0, 50], [1, 55]], -.6167903409, 2.947037200, [0, 0, 1/5]] 50700551 Comment: This probably equals, arcsinh(--------) 48600000 [`2F1(1,1;2;-2/5)`, .8411805915530323262614835255424802252787084382833586663668\ 556461585021876110287578938115512367297905, 25/4*(1+n)/(n+2)-15*(2*n+3)/(n+2)*N +N^2, [[0, 25/2], [1, 15]], .7699881947, 6.260379016, [0, 0, 2/5]] 6841 Comment: This probably equals, arctanh(----) 9966 [`2F1(1,1;2;-3/5)`, .7833393820762259227515617185805701078348317480204134007489\ 868950100031303357106369790935048267940864, 25/9*(1+n)/(n+2)-65/9*(2*n+3)/(n+2) *N+N^2, [[0, 50/9], [1, 65/9]], 1.580783933, 79.05741496, [0, 0, 3/5]] 1/3 2/3 16 5 5 5 Comment: This probably equals, arcsinh(------- - ------) 25 64 [`2F1(1,1;11/5;-1/5)`, .9190463737459224204819127432201477544462533032988486885\ 290884328370854332637958029431240343765426484, 25*(10*n+21)*(5*n+6)*(1+n)/(n+2) /(10*n+11)/(11+5*n)-20*(5*n+8)*(55*n^2+176*n+127)/(n+2)/(10*n+11)/(11+5*n)*N+N^ 2, [[0, 55], [1, 60]], 1.370989672, 56.91736073, [0, 1/5, 1/5]] [`2F1(1,1;12/5;-1/5)`, .9253051055261748891352792167580071880580095049033584004\ 296927382479610243397634766730451441026315510, 25*(5*n+11)*(5*n+7)*(1+n)/(n+2)/ (5*n+6)/(12+5*n)-5*(10*n+17)*(55*n^2+187*n+145)/(n+2)/(5*n+6)/(12+5*n)*N+N^2, [ [0, 60], [1, 65]], 1.380611401, 62.45355564, [0, 2/5, 1/5]] [`2F1(1,1;13/5;-1/5)`, .9306475132174322345753534432898146991273479597440610669\ 123830403255638159000121527013727678996433686, 25*(10*n+23)*(5*n+8)*(1+n)/(n+2) /(10*n+13)/(13+5*n)-20*(5*n+9)*(55*n^2+198*n+164)/(n+2)/(10*n+13)/(13+5*n)*N+N^ 2, [[0, 65], [1, 70]], 1.380949177, 62.66754146, [0, 3/5, 1/5]] [`2F1(1,1;14/5;-1/5)`, .9352634781105252679257852685805993274335013763284317717\ 246248599757244116105779489523820225198643288, 25*(5*n+12)*(5*n+9)*(1+n)/(n+2)/ (5*n+7)/(14+5*n)-5*(10*n+19)*(55*n^2+209*n+184)/(n+2)/(5*n+7)/(14+5*n)*N+N^2, [ [0, 70], [1, 75]], 1.394868638, 72.97068719, [0, 4/5, 1/5]] [`2F1(1,6/5;12/5;-1/5)`, .91131086128452211118489304891073950892637817514197416\ 55303416783487743848853631729643561057579039464, 5*(5*n+11)*(5*n+6)/(12+5*n)/(n +2)-11*(5*n+11)*(10*n+17)/(12+5*n)/(n+2)*N+N^2, [[0, 60], [1, 66]], .6547779375 , 7.490645329, [1/5, 1/5, 1/5]] [`2F1(1,6/5;3;-1/5)`, .92781099901479965732640293930190867514560622856434762836\ 33723093645861318066645787351735290500870716, (5*n+6)*(5+2*n)*(5*n+9)/(n+2)/(2* n+3)/(3+n)-2*(110*n^2+440*n+411)/(3+n)/(2*n+3)*N+N^2, [[0, 75], [1, 81]], -.298\ 9113272, 3.473790734, [1/5, 4/5, 1/5]] [`2F1(1,6/5;3;-2/5)`, .86813854626107831921755924385238100076766108325287134060\ 89176645670386733139248530162808018118115085, 1/4*(5*n+6)*(5+2*n)*(5*n+9)/(n+2) /(2*n+3)/(3+n)-3/2*(40*n^2+160*n+149)/(3+n)/(2*n+3)*N+N^2, [[0, 75/4], [1, 87/4 ]], 1.090918752, 10.52899060, [1/5, 4/5, 2/5]] Comment: This probably equals, 125/8 // 1/2 \ \ || 5 | Pi 1/2 1/2 1/2 Pi | (1/5) ||- ---- - 1/4| cos(----) - 1/4 2 (5 - 5 ) sin(----)| 12005 + \\ 4 / 5 5 / / / 1/2 \ \ | 1/2 1/2 1/2 Pi | 5 | Pi | 125/8 I |1/4 2 (5 - 5 ) cos(----) + |- ---- - 1/4| sin(----)| \ 5 \ 4 / 5 / (1/5) 12005 + 825/8 [`2F1(1,6/5;3;-1)`, .7362359175968965215932495630063475255218643912999237939535\ 107873312495561880497452905308782082395589, 1/25*(5*n+6)*(5+2*n)*(5*n+9)/(n+2)/ (2*n+3)/(3+n)-6/5*(10*n^2+40*n+37)/(3+n)/(2*n+3)*N+N^2, [[0, 3], [1, 21/5]], 1.\ 315579452, 7.884053796, [1/5, 4/5, 1]] Comment: This probably equals, // 1/2 \ \ || 5 | Pi 1/2 1/2 1/2 Pi | (4/5) 25 ||- ---- - 1/8| cos(----) - 1/8 2 (5 - 5 ) sin(----)| 2 + \\ 8 / 5 5 / / / 1/2 \ \ | 1/2 1/2 1/2 Pi | 5 | Pi | (4/5) 25 I |1/8 2 (5 - 5 ) cos(----) + |- ---- - 1/8| sin(----)| 2 \ 5 \ 8 / 5 / + 45/2 [`2F1(1,7/5;14/5;-1/5)`, .91107655008701731203897841207580123658234688320937826\ 40025945138078554900426147315969809226060531073, 5*(5*n+12)*(5*n+7)/(14+5*n)/(n +2)-11*(5*n+12)*(10*n+19)/(14+5*n)/(n+2)*N+N^2, [[0, 70], [1, 77]], .6576533497 , 7.516851990, [2/5, 2/5, 1/5]] [`2F1(1,7/5;3;-1/5)`, .91653713961926561953945631431759347560423081586475080332\ 87283500443350800769323892544133352396538561, (5*n+7)*(5+2*n)*(5*n+8)/(n+2)/(2* n+3)/(3+n)-4*(55*n^2+220*n+206)/(3+n)/(2*n+3)*N+N^2, [[0, 75], [1, 82]], -.2676\ 688244, 3.535907189, [2/5, 3/5, 1/5]] [`2F1(1,7/5;3;-2/5)`, .84868516955942463949822854507601910663715389325038513699\ 37443044157875378384374691658820469178506310, 1/4*(5*n+7)*(5+2*n)*(5*n+8)/(n+2) /(2*n+3)/(3+n)-1/2*(120*n^2+480*n+449)/(3+n)/(2*n+3)*N+N^2, [[0, 75/4], [1, 89/ 4]], 1.121112285, 11.25071075, [2/5, 3/5, 2/5]] Comment: This probably equals, // 1/2 \ \ 125 || 5 | Pi 1/2 1/2 1/2 Pi | (1/5) --- ||- ---- - 1/4| cos(----) - 1/4 2 (5 - 5 ) sin(----)| 8575 12 \\ 4 / 5 5 / / / 1/2 \ \ 125 | 1/2 1/2 1/2 Pi | 5 | Pi | + --- I |1/4 2 (5 - 5 ) cos(----) + |- ---- - 1/4| sin(----)| 12 \ 5 \ 4 / 5 / (1/5) 775 8575 + --- 12 [`2F1(1,7/5;3;-1)`, .7023619457466826471061683224462896776559704751821529830207\ 896491878860233161810196568655429214017447, 1/25*(5*n+7)*(5+2*n)*(5*n+8)/(n+2)/ (2*n+3)/(3+n)-4/5*(15*n^2+60*n+56)/(3+n)/(2*n+3)*N+N^2, [[0, 3], [1, 22/5]], 1.\ 339406088, 8.327786964, [2/5, 3/5, 1]] Comment: This probably equals, // 1/2 \ \ || 5 | Pi 1/2 1/2 1/2 Pi | (3/5) 25/3 ||- ---- - 1/4| cos(----) - 1/4 2 (5 - 5 ) sin(----)| 2 \\ 4 / 5 5 / + / / 1/2 \ \ | 1/2 1/2 1/2 Pi | 5 | Pi | (3/5) 25/3 I |1/4 2 (5 - 5 ) cos(----) + |- ---- - 1/4| sin(----)| 2 \ 5 \ 4 / 5 / + 40/3 [`2F1(1,8/5;3;-1/5)`, .90546730571455837166980192774104078070336968608949015558\ 02898625560101857380340467898145239907544768, (5*n+7)*(5+2*n)*(5*n+8)/(n+2)/(2* n+3)/(3+n)-2*(110*n^2+440*n+413)/(3+n)/(2*n+3)*N+N^2, [[0, 75], [1, 83]], -.264\ 5334398, 3.542263847, [3/5, 2/5, 1/5]] [`2F1(1,8/5;3;-2/5)`, .82987729902484405393960999361664575351363318244392288186\ 51960732375796221411305604623851464830613355, 1/4*(5*n+7)*(5+2*n)*(5*n+8)/(n+2) /(2*n+3)/(3+n)-1/2*(120*n^2+480*n+451)/(3+n)/(2*n+3)*N+N^2, [[0, 75/4], [1, 91/ 4]], 1.119254900, 11.20346937, [3/5, 2/5, 2/5]] Comment: This probably equals, // 1/2 \ \ 125 || 5 | Pi 1/2 1/2 1/2 Pi | (1/5) --- ||- ---- - 1/4| cos(----) - 1/4 2 (5 - 5 ) sin(----)| 6125 12 \\ 4 / 5 5 / / / 1/2 \ \ 125 | 1/2 1/2 1/2 Pi | 5 | Pi | + --- I |1/4 2 (5 - 5 ) cos(----) + |- ---- - 1/4| sin(----)| 12 \ 5 \ 4 / 5 / (1/5) 725 6125 + --- 12 [`2F1(1,8/5;3;-1)`, .6707674102258811718833169064196655580544248900548443207849\ 350334955334925539930829137183730336708448, 1/25*(5*n+7)*(5+2*n)*(5*n+8)/(n+2)/ (2*n+3)/(3+n)-2/5*(30*n^2+120*n+113)/(3+n)/(2*n+3)*N+N^2, [[0, 3], [1, 23/5]], 1.344904473, 8.437372172, [3/5, 2/5, 1]] Comment: This probably equals, // 1/2 \ \ || 5 | Pi 1/2 1/2 1/2 Pi | (2/5) 25/3 ||- ---- - 1/4| cos(----) - 1/4 2 (5 - 5 ) sin(----)| 2 \\ 4 / 5 5 / + / / 1/2 \ \ | 1/2 1/2 1/2 Pi | 5 | Pi | (2/5) 25/3 I |1/4 2 (5 - 5 ) cos(----) + |- ---- - 1/4| sin(----)| 2 \ 5 \ 4 / 5 / + 35/3 [`2F1(1,8/5;16/5;-1/5)`, .91088694358067085557442386132892791698858543338838262\ 56985837116249252566002869076288789154040549398, 5*(5*n+13)*(5*n+8)/(16+5*n)/(n +2)-11*(5*n+13)*(10*n+21)/(16+5*n)/(n+2)*N+N^2, [[0, 80], [1, 88]], .6430091915 , 7.385262063, [3/5, 3/5, 1/5]] [`2F1(1,9/5;3;-1/5)`, .89459708229746827684494500452205196273857700592994809051\ 95339409986707208962063249757108386023797756, (5*n+6)*(5+2*n)*(5*n+9)/(n+2)/(2* n+3)/(3+n)-4*(55*n^2+220*n+207)/(3+n)/(2*n+3)*N+N^2, [[0, 75], [1, 84]], -.2932\ 345268, 3.484914599, [4/5, 1/5, 1/5]] [`2F1(1,9/5;3;-2/5)`, .81168939647899464973713909321545217751423357791528950745\ 07274199644825837851583833069818198891697023, 1/4*(5*n+6)*(5+2*n)*(5*n+9)/(n+2) /(2*n+3)/(3+n)-3/2*(40*n^2+160*n+151)/(3+n)/(2*n+3)*N+N^2, [[0, 75/4], [1, 93/4 ]], 1.091718050, 10.54690113, [4/5, 1/5, 2/5]] Comment: This probably equals, 125/8 // 1/2 \ \ || 5 | Pi 1/2 1/2 1/2 Pi | (1/5) ||- ---- - 1/4| cos(----) - 1/4 2 (5 - 5 ) sin(----)| 4375 + \\ 4 / 5 5 / / / 1/2 \ \ | 1/2 1/2 1/2 Pi | 5 | Pi | 125/8 I |1/4 2 (5 - 5 ) cos(----) + |- ---- - 1/4| sin(----)| \ 5 \ 4 / 5 / (1/5) 4375 + 675/8 [`2F1(1,9/5;3;-1)`, .6412705625370624150171631652759051319518638862775311810786\ 110188299599218310867898565860873561440761, 1/25*(5*n+6)*(5+2*n)*(5*n+9)/(n+2)/ (2*n+3)/(3+n)-12/5*(5*n^2+20*n+19)/(3+n)/(2*n+3)*N+N^2, [[0, 3], [1, 24/5]], 1.\ 314920003, 7.872444110, [4/5, 1/5, 1]] Comment: This probably equals, // 1/2 \ \ || 5 | Pi 1/2 1/2 1/2 Pi | (1/5) 25 ||- ---- - 1/8| cos(----) - 1/8 2 (5 - 5 ) sin(----)| 2 + \\ 8 / 5 5 / / / 1/2 \ \ | 1/2 1/2 1/2 Pi | 5 | Pi | (1/5) 25 I |1/8 2 (5 - 5 ) cos(----) + |- ---- - 1/8| sin(----)| 2 \ 5 \ 8 / 5 / + 15 [`2F1(1,9/5;18/5;-1/5)`, .91073036736539899708633669083910494685675401190843214\ 84022237428283176141990133711483159687724210214, 5*(5*n+14)*(5*n+9)/(18+5*n)/(n +2)-11*(5*n+14)*(10*n+23)/(18+5*n)/(n+2)*N+N^2, [[0, 90], [1, 99]], .6339963977 , 7.306540540, [4/5, 4/5, 1/5]] ----------------------- denominator , 6 [`2F1(1,1;2;-1/6)`, .9249040789635498257572523103748647414845642243931388956240\ 948408401516853825935118490765006663245040, 36*(1+n)/(n+2)-78*(2*n+3)/(n+2)*N+N ^2, [[0, 72], [1, 78]], -.8020126341, 2.873228774, [0, 0, 1/6]] 11664504865 Comment: This probably equals, arcsinh(-----------) 10978063488 [`2F1(1,1;7/3;-1/6)`, .93498119832813669787765320733258688883295163047181587047\ 28736244556704979959961436209076185463182384, 36*(3*n+4)*(6*n+13)*(1+n)/(n+2)/( 6*n+7)/(7+3*n)-12*(3*n+5)*(78*n^2+260*n+197)/(n+2)/(6*n+7)/(7+3*n)*N+N^2, [[0, 84], [1, 90]], 1.005734906, 14.18634203, [0, 1/3, 1/6]] [`2F1(1,1;5/2;-1/6)`, .93905290627777000183041981424266976985345656773905150282\ 72853135266560825888149851685903376517626100, 36*(2*n+3)*(4*n+9)*(1+n)/(n+2)/(4 *n+5)/(5+2*n)-12*(4*n+7)*(26*n^2+91*n+73)/(n+2)/(4*n+5)/(5+2*n)*N+N^2, [[0, 90] , [1, 96]], .8882214401, 11.29527271, [0, 1/2, 1/6]] [`2F1(1,1;8/3;-1/6)`, .94263674859197329710288633744318319294668882840518290044\ 37198892329185699990088712760955575633865086, 36*(3*n+5)*(3*n+7)*(1+n)/(n+2)/(3 *n+4)/(8+3*n)-6*(6*n+11)*(39*n^2+143*n+121)/(n+2)/(3*n+4)/(8+3*n)*N+N^2, [[0, 96], [1, 102]], .9918684720, 13.77044399, [0, 2/3, 1/6]] [`2F1(1,7/6;7/3;-1/6)`, .924720997180507664434549800095725412742948926114417972\ 2121114257737359877336695082170834510916381588, 3*(6*n+13)*(6*n+7)/(7+3*n)/(n+2 )-26*(6*n+13)*(3*n+5)/(7+3*n)/(n+2)*N+N^2, [[0, 84], [1, 91]], .2433693041, 5.3\ 32220020, [1/6, 1/6, 1/6]] [`2F1(1,7/6;7/3;-1/3)`, .862451764609170901338637681396050604420604461770271131\ 5583462528000876110138585575016896221025929324, 3/4*(6*n+13)*(6*n+7)/(7+3*n)/(n +2)-7*(6*n+13)*(3*n+5)/(7+3*n)/(n+2)*N+N^2, [[0, 21], [1, 49/2]], .9362338022, 8.793352810, [1/6, 1/6, 1/3]] [`2F1(1,7/6;7/3;-1/2)`, .809822073441813092853973560887639404650696480416690546\ 7657782251923324014819263465390825461156890366, 1/3*(6*n+13)*(6*n+7)/(7+3*n)/(n +2)-10/3*(6*n+13)*(3*n+5)/(7+3*n)/(n+2)*N+N^2, [[0, 28/3], [1, 35/3]], 1.340463\ 855, 17.71444263, [1/6, 1/6, 1/2]] [`2F1(1,7/6;7/3;-2/3)`, .764578991521671819175512767351509323591847509615408242\ 7267692513941121200287186393066653285367129241, 3/16*(6*n+13)*(6*n+7)/(7+3*n)/( n+2)-2*(6*n+13)*(3*n+5)/(7+3*n)/(n+2)*N+N^2, [[0, 21/4], [1, 7]], 1.626897746, 132.8070364, [1/6, 1/6, 2/3]] [`2F1(1,7/6;7/3;-1)`, .69039261685143641514661590335262537532286808817305029547\ 34606466573368875026137755885607329901012780, 1/12*(6*n+13)*(6*n+7)/(7+3*n)/(n+ 2)-(6*n+13)*(3*n+5)/(7+3*n)/(n+2)*N+N^2, [[0, 7/3], [1, 7/2]], 1.209312178, 6.3\ 70204983, [1/6, 1/6, 1]] [`2F1(1,7/6;5/2;-1/3)`, .870519328386554292369245767715012690249587116187439176\ 3637882009773253808666966909800234336524724365, (3*n+4)*(6*n+7)*(4*n+9)/(n+2)/( 4*n+5)/(5+2*n)-2*(4*n+7)*(42*n^2+147*n+118)/(n+2)/(4*n+5)/(5+2*n)*N+N^2, [[0, 45/2], [1, 26]], .8091674022, 7.254605986, [1/6, 1/3, 1/3]] [`2F1(1,7/6;3;-1/6)`, .94048257152458193483477650575742014467502993459245901542\ 78321258999510815304525469389336927280676206, (5+2*n)*(6*n+7)*(6*n+11)/(n+2)/(2 *n+3)/(3+n)-2*(156*n^2+624*n+583)/(3+n)/(2*n+3)*N+N^2, [[0, 108], [1, 115]], .9\ 159641890, 11.86617348, [1/6, 5/6, 1/6]] [`2F1(1,4/3;5/2;-1/6)`, .919909240648268753900547476433300579958165004297461225\ 0873106324008123383319352011630982884408923605, 4*(3*n+4)*(6*n+7)*(4*n+9)/(n+2) /(4*n+5)/(5+2*n)-2*(4*n+7)*(156*n^2+546*n+439)/(n+2)/(4*n+5)/(5+2*n)*N+N^2, [[0 , 90], [1, 98]], .8090320820, 9.931389234, [1/3, 1/6, 1/6]] [`2F1(1,4/3;8/3;-1/6)`, .924571242238351368565129519638654315828010142448212504\ 7846487128665664904451561188391070580927371961, 12*(3*n+7)*(3*n+4)/(8+3*n)/(n+2 )-26*(3*n+7)*(6*n+11)/(8+3*n)/(n+2)*N+N^2, [[0, 96], [1, 104]], .8714338787e-1, 4.727571531, [1/3, 1/3, 1/6]] [`2F1(1,4/3;8/3;-1/2)`, .808917050867206638920872694658582314025408585660229163\ 3766888959877906955929260581611249927751274656, 4/3*(3*n+7)*(3*n+4)/(8+3*n)/(n+ 2)-10/3*(3*n+7)*(6*n+11)/(8+3*n)/(n+2)*N+N^2, [[0, 32/3], [1, 40/3]], 1.1851923\ 37, 11.07208450, [1/3, 1/3, 1/2]] [`2F1(1,4/3;3;-1/6)`, .93241289784730991472189337555319572537192066337916223122\ 92880430909602878324849431235332164520747931, 4*(3*n+5)*(5+2*n)*(3*n+4)/(n+2)/( 2*n+3)/(3+n)-8*(39*n^2+156*n+146)/(3+n)/(2*n+3)*N+N^2, [[0, 108], [1, 116]], -.\ 8939380806, 2.761255030, [1/3, 2/3, 1/6]] (1/3) Comment: This probably equals, -54 294 + 360 [`2F1(1,4/3;3;-1/2)`, .82665490423986107145009695204406713569030237697462894815\ 00714968834032544712420410877951321966752402, 4/9*(3*n+5)*(5+2*n)*(3*n+4)/(n+2) /(2*n+3)/(3+n)-8/3*(15*n^2+60*n+56)/(3+n)/(2*n+3)*N+N^2, [[0, 12], [1, 44/3]], .2087477189, 3.297189567, [1/3, 2/3, 1/2]] (1/3) Comment: This probably equals, -18 18 + 48 [`2F1(1,3/2;5/2;-1/3)`, .837902860946019672326647840610299204721701883582667409\ 9355177907534456849314075234214375530457292401, 9*(2*n+3)*(4*n+9)*(1+n)/(n+2)/( 4*n+5)/(5+2*n)-3*(4*n+7)*(28*n^2+98*n+79)/(n+2)/(4*n+5)/(5+2*n)*N+N^2, [[0, 45/ 2], [1, 27]], .8888411574, 8.148705489, [1/2, 0, 1/3]] [`2F1(1,3/2;8/3;-1/6)`, .915726743235907508335203819722928590540234577499277648\ 7658266588424852135745415272708140582415900550, 9*(2*n+3)*(3*n+7)*(6*n+7)/(n+2) /(3*n+4)/(8+3*n)-3*(6*n+11)*(78*n^2+286*n+243)/(n+2)/(3*n+4)/(8+3*n)*N+N^2, [[0 , 96], [1, 105]], 1.166212477, 21.80948059, [1/2, 1/6, 1/6]] [`2F1(1,3/2;3;-1/6)`, .92444647642270891363356306777616043015019326119936561385\ 49889051383658634858701727826764829307086876, 9*(5+2*n)*(2*n+3)/(3+n)/(n+2)-78* (5+2*n)/(3+n)*N+N^2, [[0, 108], [1, 117]], -1.790465266, 2.000795119, [1/2, 1/2 , 1/6]] 1/2 Comment: This probably equals, 312 - 48 42 [`2F1(1,3/2;3;-1/3)`, .86156123669388991068257560771812638674534781710172985332\ 12649863072151883775982201049830356520683676, 9/4*(5+2*n)*(2*n+3)/(3+n)/(n+2)-\ 21*(5+2*n)/(3+n)*N+N^2, [[0, 27], [1, 63/2]], -1.097318085, 2.000983189, [1/2, 1/2, 1/3]] 1/2 Comment: This probably equals, 84 - 48 3 [`2F1(1,3/2;3;-2/3)`, .76209992275549868892440760130560233500246976825045504047\ 45574033191014483781257988842757388479588925, 9/16*(5+2*n)*(2*n+3)/(3+n)/(n+2)-\ 6*(5+2*n)/(3+n)*N+N^2, [[0, 27/4], [1, 9]], -.4041709048, 2.001255203, [1/2, 1/ 2, 2/3]] 1/2 Comment: This probably equals, 24 - 6 15 [`2F1(1,3/2;3;-5/6)`, .72184626309895610791702354892920253087192815348739937090\ 89748584967997385459084256016174023399611953, 9/25*(5+2*n)*(2*n+3)/(3+n)/(n+2)-\ 102/25*(5+2*n)/(3+n)*N+N^2, [[0, 108/25], [1, 153/25]], 1.428410559, 13.3501020\ 4, [1/2, 1/2, 5/6]] 1/2 408 48 66 Comment: This probably equals, --- - -------- 25 25 [`2F1(1,3/2;10/3;-1/6)`, .93150584574113625182199271891731253359992269346037894\ 96742047539090228111414824161595353916946261523, 9*(6*n+11)*(3*n+8)*(2*n+3)/(n+ 2)/(3*n+5)/(10+3*n)-6*(6*n+13)*(39*n^2+169*n+173)/(n+2)/(3*n+5)/(10+3*n)*N+N^2, [[0, 120], [1, 129]], 1.134067020, 19.69006644, [1/2, 5/6, 1/6]] [`2F1(1,5/3;3;-1/6)`, .91658172655553206509407399260176077152260146493639700222\ 17474528280474035754909433740144210399665896, 4*(3*n+5)*(5+2*n)*(3*n+4)/(n+2)/( 2*n+3)/(3+n)-4*(78*n^2+312*n+293)/(3+n)/(2*n+3)*N+N^2, [[0, 108], [1, 118]], -.\ 8967543315, 2.757962197, [2/3, 1/3, 1/6]] (1/3) Comment: This probably equals, -54 252 + 342 [`2F1(1,5/3;3;-1/2)`, .79028726808005275891048037017162767902735085163448140443\ 41217336357252351282905842461062028869863602, 4/9*(3*n+5)*(5+2*n)*(3*n+4)/(n+2) /(2*n+3)/(3+n)-4/3*(30*n^2+120*n+113)/(3+n)/(2*n+3)*N+N^2, [[0, 12], [1, 46/3]] , .2075351902, 3.294316967, [2/3, 1/3, 1/2]] (1/3) Comment: This probably equals, -18 12 + 42 [`2F1(1,5/3;10/3;-1/6)`, .92434092863778012788681608684627501294112134948675181\ 33411911025988027534702197190039772797479561746, 12*(3*n+8)*(3*n+5)/(10+3*n)/(n +2)-26*(3*n+8)*(6*n+13)/(10+3*n)/(n+2)*N+N^2, [[0, 120], [1, 130]], .6291735161\ e-1, 4.645877200, [2/3, 2/3, 1/6]] [`2F1(1,5/3;10/3;-1/2)`, .80752798620235320006341811926673855389889434287557981\ 43932443167735658562670474453277802628225193416, 4/3*(3*n+8)*(3*n+5)/(10+3*n)/( n+2)-10/3*(3*n+8)*(6*n+13)/(10+3*n)/(n+2)*N+N^2, [[0, 40/3], [1, 50/3]], 1.1590\ 92625, 10.41560210, [2/3, 2/3, 1/2]] [`2F1(1,5/3;7/2;-1/6)`, .927695104655120087676395649165448115519446159371778448\ 5665102310930123577002407943224896442029188576, 4*(3*n+5)*(4*n+11)*(6*n+11)/(n+ 2)/(4*n+7)/(7+2*n)-12*(4*n+9)*(26*n^2+117*n+125)/(n+2)/(4*n+7)/(7+2*n)*N+N^2, [ [0, 126], [1, 136]], .7856756434, 9.589856403, [2/3, 5/6, 1/6]] [`2F1(1,11/6;3;-1/6)`, .9088170943930946162926760187186500439144144072856250725\ 065982071881802294871531552022041745719874322, (5+2*n)*(6*n+7)*(6*n+11)/(n+2)/( 2*n+3)/(3+n)-2*(156*n^2+624*n+587)/(3+n)/(2*n+3)*N+N^2, [[0, 108], [1, 119]], .\ 9187370218, 11.92642226, [5/6, 1/6, 1/6]] [`2F1(1,11/6;7/2;-1/3)`, .85520663566304341417841206434782625041618752375233546\ 44315868005343896591229485459132039151427865288, (3*n+5)*(4*n+11)*(6*n+11)/(n+2 )/(4*n+7)/(7+2*n)-3*(4*n+9)*(28*n^2+126*n+135)/(n+2)/(4*n+7)/(7+2*n)*N+N^2, [[0 , 63/2], [1, 37]], .7923646336, 7.090531307, [5/6, 2/3, 1/3]] [`2F1(1,11/6;11/3;-1/6)`, .9242504774855014818119206946153400933374938646704079\ 985128191533000200697372134466037984201474627217, 3*(6*n+17)*(6*n+11)/(11+3*n)/ (n+2)-26*(6*n+17)*(3*n+7)/(11+3*n)/(n+2)*N+N^2, [[0, 132], [1, 143]], .20556957\ 45, 5.172163732, [5/6, 5/6, 1/6]] [`2F1(1,11/6;11/3;-1/3)`, .8609261305450455908331696655600510322263761847538589\ 769750228028567678834404246036193857186638000699, 3/4*(6*n+17)*(6*n+11)/(11+3*n )/(n+2)-7*(6*n+17)*(3*n+7)/(11+3*n)/(n+2)*N+N^2, [[0, 33], [1, 77/2]], .9027033\ 470, 8.327268346, [5/6, 5/6, 1/3]] [`2F1(1,11/6;11/3;-1/2)`, .8069834967402765279660315488869961947881865741365586\ 680027738615140532834973183130936986185939749618, 1/3*(6*n+17)*(6*n+11)/(11+3*n )/(n+2)-10/3*(6*n+17)*(3*n+7)/(11+3*n)/(n+2)*N+N^2, [[0, 44/3], [1, 55/3]], 1.3\ 05931440, 15.62916525, [5/6, 5/6, 1/2]] [`2F1(1,11/6;11/3;-2/3)`, .7603377722106513993794610320400681309888832018710300\ 949368783381162035541427300154957748980008811092, 3/16*(6*n+17)*(6*n+11)/(11+3* n)/(n+2)-2*(6*n+17)*(3*n+7)/(11+3*n)/(n+2)*N+N^2, [[0, 33/4], [1, 11]], 1.59179\ 6634, 62.36273317, [5/6, 5/6, 2/3]] [`2F1(1,11/6;11/3;-1)`, .683388237618135942265119789155036359929670931770349318\ 4839467132359700066879055727176852396020446379, 1/12*(6*n+17)*(6*n+11)/(11+3*n) /(n+2)-(6*n+17)*(3*n+7)/(11+3*n)/(n+2)*N+N^2, [[0, 11/3], [1, 11/2]], 1.1771819\ 38, 6.020668836, [5/6, 5/6, 1]] ----------------------- denominator , 7 [`2F1(1,1;2;-1/7)`, .9347197483716583620244053465194498221259097144923320173185\ 491497982070220991041532095203768096917494, 49*(1+n)/(n+2)-105*(2*n+3)/(n+2)*N+ N^2, [[0, 98], [1, 105]], -.9500356851, 2.828442275, [0, 0, 1/7]] 3719823438255 Comment: This probably equals, arcsinh(-------------) 3454189699072 [`2F1(1,1;2;-2/7)`, .8796004989831712718979820564065508788026373512227320025349\ 948106809305328980203308585055386385742770, 49/4*(1+n)/(n+2)-28*(2*n+3)/(n+2)*N +N^2, [[0, 49/2], [1, 28]], .4262207719, 5.081621061, [0, 0, 2/7]] 1979713 Comment: This probably equals, arctanh(-------) 2803256 [`2F1(1,1;2;-3/7)`, .8322415358570422174628236595627637819160391044427941710058\ 547524033164094923274957127903151706957998, 49/9*(1+n)/(n+2)-119/9*(2*n+3)/(n+2 )*N+N^2, [[0, 98/9], [1, 119/9]], 1.240723043, 14.58414816, [0, 0, 3/7]] 1/3 2/3 50 490 7 490 Comment: This probably equals, arcsinh(--------- - --------) 343 2000 [`2F1(1,1;15/7;-1/7)`, .9388294079387791900498667826163611580435756876174860152\ 951436472184556924594249685206106940652833127, 49*(7*n+8)*(14*n+29)*(1+n)/(n+2) /(14*n+15)/(15+7*n)-28*(7*n+11)*(105*n^2+330*n+233)/(n+2)/(14*n+15)/(15+7*n)*N+ N^2, [[0, 105], [1, 112]], 1.232280693, 30.64609998, [0, 1/7, 1/7]] [`2F1(1,1;16/7;-1/7)`, .9424440831768353392846144066164324946410091899530943423\ 719328673704292948090334169278910630138360357, 49*(7*n+15)*(7*n+9)*(1+n)/(n+2)/ (7*n+8)/(16+7*n)-7*(14*n+23)*(105*n^2+345*n+257)/(n+2)/(7*n+8)/(16+7*n)*N+N^2, [[0, 112], [1, 119]], 1.216001903, 28.55147537, [0, 2/7, 1/7]] [`2F1(1,1;17/7;-1/7)`, .9456490072483507563459369863471524369862059276358567134\ 230680160407372924157926948027221283994263103, 49*(7*n+10)*(14*n+31)*(1+n)/(n+2 )/(14*n+17)/(17+7*n)-84*(7*n+12)*(35*n^2+120*n+94)/(n+2)/(14*n+17)/(17+7*n)*N+N ^2, [[0, 119], [1, 126]], 1.238878694, 31.58528845, [0, 3/7, 1/7]] [`2F1(1,1;18/7;-1/7)`, .9485107831952411811469511277456574370789959346348907183\ 390180580993789839828652905129495723581248457, 49*(7*n+11)*(7*n+16)*(1+n)/(n+2) /(7*n+9)/(18+7*n)-7*(14*n+25)*(105*n^2+375*n+308)/(n+2)/(7*n+9)/(18+7*n)*N+N^2, [[0, 126], [1, 133]], 1.227178202, 29.95722664, [0, 4/7, 1/7]] [`2F1(1,1;19/7;-1/7)`, .9510822312635294218218624649092392592233901265200248253\ 719151888306109900529123483031072917869785529, 49*(14*n+33)*(7*n+12)*(1+n)/(n+2 )/(14*n+19)/(19+7*n)-140*(7*n+13)*(21*n^2+78*n+67)/(n+2)/(14*n+19)/(19+7*n)*N+N ^2, [[0, 133], [1, 140]], 1.214559253, 28.37957542, [0, 5/7, 1/7]] [`2F1(1,1;20/7;-1/7)`, .9534057935174174205458499435995444434682042820192321176\ 999692606502427958685397074328958296886551606, 49*(7*n+13)*(7*n+17)*(1+n)/(n+2) /(7*n+10)/(20+7*n)-21*(14*n+27)*(35*n^2+135*n+121)/(n+2)/(7*n+10)/(20+7*n)*N+N^ 2, [[0, 140], [1, 147]], 1.234010366, 30.88686499, [0, 6/7, 1/7]] [`2F1(1,8/7;16/7;-1/7)`, .93459900758455854472730584222948901216357702491178716\ 92786379810685685292200120110310769794988055211, 7*(7*n+15)*(7*n+8)/(16+7*n)/(n +2)-15*(7*n+15)*(14*n+23)/(16+7*n)/(n+2)*N+N^2, [[0, 112], [1, 120]], .56842325\ 80, 7.677292630, [1/7, 1/7, 1/7]] [`2F1(1,8/7;3;-1/7)`, .94940256897221921505456545665943788609315008708166010816\ 05431706855405265712733059671215326636942966, (7*n+8)*(5+2*n)*(7*n+13)/(n+2)/(2 *n+3)/(3+n)-10*(42*n^2+168*n+157)/(3+n)/(2*n+3)*N+N^2, [[0, 147], [1, 155]], -.\ 3119824494, 3.850263759, [1/7, 6/7, 1/7]] [`2F1(1,8/7;3;-2/7)`, .90531276328097521348259032083772784172686570907075581874\ 80755574700685735456017757416651810394596062, 1/4*(7*n+8)*(5+2*n)*(7*n+13)/(n+2 )/(2*n+3)/(3+n)-1/2*(224*n^2+896*n+835)/(3+n)/(2*n+3)*N+N^2, [[0, 147/4], [1, 163/4]], 1.072490582, 12.48815004, [1/7, 6/7, 2/7]] [`2F1(1,8/7;3;-1)`, .7463603007123820612570194749732958684521012122407130661222\ 010414421994010902806594776964662261645617, 1/49*(7*n+8)*(5+2*n)*(7*n+13)/(n+2) /(2*n+3)/(3+n)-2/7*(42*n^2+168*n+155)/(3+n)/(2*n+3)*N+N^2, [[0, 3], [1, 29/7]], 1.632461099, 27.05964127, [1/7, 6/7, 1]] [`2F1(1,9/7;18/7;-1/7)`, .93449760261342181075779528760844350634777797258241229\ 96223980369607672033294213649490714200654874116, 7*(7*n+16)*(7*n+9)/(18+7*n)/(n +2)-15*(7*n+16)*(14*n+25)/(18+7*n)/(n+2)*N+N^2, [[0, 126], [1, 135]], .55795193\ 48, 7.587592313, [2/7, 2/7, 1/7]] [`2F1(1,9/7;3;-1/7)`, .94334746742995956721997413015159188374740657509764701082\ 52237780359384601012904954252819946322541847, (5+2*n)*(7*n+9)*(7*n+12)/(n+2)/(2 *n+3)/(3+n)-12*(35*n^2+140*n+131)/(3+n)/(2*n+3)*N+N^2, [[0, 147], [1, 156]], -.\ 3380001795, 3.794367899, [2/7, 5/7, 1/7]] [`2F1(1,9/7;3;-2/7)`, .89442007214150112188769401576081493124075368100180232112\ 03908756865211712034800170712074664633483425, 1/4*(5+2*n)*(7*n+9)*(7*n+12)/(n+2 )/(2*n+3)/(3+n)-1/2*(224*n^2+896*n+837)/(3+n)/(2*n+3)*N+N^2, [[0, 147/4], [1, 165/4]], 1.047374178, 11.81868894, [2/7, 5/7, 2/7]] [`2F1(1,9/7;3;-1)`, .7214270222502965490624202157970774220638318300009525833358\ 404554976981261595768377280772507245327852, 1/49*(5+2*n)*(7*n+9)*(7*n+12)/(n+2) /(2*n+3)/(3+n)-12/7*(7*n^2+28*n+26)/(3+n)/(2*n+3)*N+N^2, [[0, 3], [1, 30/7]], 1\ .606281864, 22.53211549, [2/7, 5/7, 1]] [`2F1(1,10/7;20/7;-1/7)`, .9344112342649007246459906173170500823768754656928482\ 711762093785369879446012159033830731230322691764, 7*(7*n+17)*(7*n+10)/(20+7*n)/ (n+2)-15*(7*n+17)*(14*n+27)/(20+7*n)/(n+2)*N+N^2, [[0, 140], [1, 150]], .565430\ 1388, 7.651436978, [3/7, 3/7, 1/7]] [`2F1(1,10/7;3;-1/7)`, .9373500041596823183782095969816161634995514204860726958\ 585845938994255983167590544323060615104433805, (7*n+11)*(5+2*n)*(7*n+10)/(n+2)/ (2*n+3)/(3+n)-2*(210*n^2+840*n+787)/(3+n)/(2*n+3)*N+N^2, [[0, 147], [1, 157]], -.3213164350, 3.830022420, [3/7, 4/7, 1/7]] [`2F1(1,10/7;3;-2/7)`, .8837221563366790194604191614651839495815396729396085750\ 116960146111492179706037735023375066245314650, 1/4*(7*n+11)*(5+2*n)*(7*n+10)/(n +2)/(2*n+3)/(3+n)-1/2*(224*n^2+896*n+839)/(3+n)/(2*n+3)*N+N^2, [[0, 147/4], [1, 167/4]], 1.058827507, 12.11484413, [3/7, 4/7, 2/7]] [`2F1(1,10/7;3;-1)`, .697713305381587864134531491830827048356357231263603844255\ 2238899352079902187910292794163815882596576, 1/49*(7*n+11)*(5+2*n)*(7*n+10)/(n+ 2)/(2*n+3)/(3+n)-2/7*(42*n^2+168*n+157)/(3+n)/(2*n+3)*N+N^2, [[0, 3], [1, 31/7] ], 1.624593714, 25.51868369, [3/7, 4/7, 1]] [`2F1(1,11/7;3;-1/7)`, .9314095224179462574695350407981547577238256519727057930\ 400586208202276925258070344992290330603431769, (7*n+11)*(5+2*n)*(7*n+10)/(n+2)/ (2*n+3)/(3+n)-4*(105*n^2+420*n+394)/(3+n)/(2*n+3)*N+N^2, [[0, 147], [1, 158]], -.3237923480, 3.824688909, [4/7, 3/7, 1/7]] [`2F1(1,11/7;3;-2/7)`, .8732148557966654047427846521648558802077828238706591395\ 448129713293819371214538242924498503607447662, 1/4*(7*n+11)*(5+2*n)*(7*n+10)/(n +2)/(2*n+3)/(3+n)-1/2*(224*n^2+896*n+841)/(3+n)/(2*n+3)*N+N^2, [[0, 147/4], [1, 169/4]], 1.062557518, 12.21452342, [4/7, 3/7, 2/7]] [`2F1(1,11/7;3;-1)`, .675148426835758255800200402700641114342315947983940093456\ 0210895772482989360185177057630710002882238, 1/49*(7*n+11)*(5+2*n)*(7*n+10)/(n+ 2)/(2*n+3)/(3+n)-4/7*(21*n^2+84*n+79)/(3+n)/(2*n+3)*N+N^2, [[0, 3], [1, 32/7]], 1.621831205, 25.01841678, [4/7, 3/7, 1]] [`2F1(1,11/7;22/7;-1/7)`, .9343367897365880541198051332653301711611274183063753\ 546111786939347726006912956446838254730772828129, 7*(7*n+18)*(7*n+11)/(22+7*n)/ (n+2)-15*(7*n+18)*(14*n+29)/(22+7*n)/(n+2)*N+N^2, [[0, 154], [1, 165]], .588572\ 2189, 7.856000030, [4/7, 4/7, 1/7]] [`2F1(1,12/7;3;-1/7)`, .9255253737639618451532909880284931355541222357067837499\ 259759212248173325238507915548471766258248497, (5+2*n)*(7*n+9)*(7*n+12)/(n+2)/( 2*n+3)/(3+n)-6*(70*n^2+280*n+263)/(3+n)/(2*n+3)*N+N^2, [[0, 147], [1, 159]], -.\ 3380001795, 3.794367899, [5/7, 2/7, 1/7]] [`2F1(1,12/7;3;-2/7)`, .8628941089143669577554802811425707650366475964425412910\ 052411640244908735491084978899647573448074819, 1/4*(5+2*n)*(7*n+9)*(7*n+12)/(n+ 2)/(2*n+3)/(3+n)-1/2*(224*n^2+896*n+843)/(3+n)/(2*n+3)*N+N^2, [[0, 147/4], [1, 171/4]], 1.048830661, 11.85554397, [5/7, 2/7, 2/7]] [`2F1(1,12/7;3;-1)`, .653666188796140679065428174592903598681363028572384095408\ 4791546457405044952536605908702249256897776, 1/49*(5+2*n)*(7*n+9)*(7*n+12)/(n+2 )/(2*n+3)/(3+n)-6/7*(14*n^2+56*n+53)/(3+n)/(2*n+3)*N+N^2, [[0, 3], [1, 33/7]], 1.607591466, 22.72229930, [5/7, 2/7, 1]] [`2F1(1,12/7;24/7;-1/7)`, .9342719598256943231887929628517562646496492725335128\ 391959637886025722734173799374181974929630740887, 7*(7*n+19)*(7*n+12)/(24+7*n)/ (n+2)-15*(7*n+19)*(14*n+31)/(24+7*n)/(n+2)*N+N^2, [[0, 168], [1, 180]], .603191\ 5082, 7.990960641, [5/7, 5/7, 1/7]] [`2F1(1,13/7;3;-1/7)`, .9196969179472310662529308510166004652075715376128215653\ 949906775248322537488646506490650397360484686, (7*n+8)*(5+2*n)*(7*n+13)/(n+2)/( 2*n+3)/(3+n)-20*(21*n^2+84*n+79)/(3+n)/(2*n+3)*N+N^2, [[0, 147], [1, 160]], -.3\ 154097917, 3.842806541, [6/7, 1/7, 1/7]] [`2F1(1,13/7;3;-2/7)`, .8527559500526599343111238392662279650604156972238806567\ 090570219185254194459312404136203667817348100, 1/4*(7*n+8)*(5+2*n)*(7*n+13)/(n+ 2)/(2*n+3)/(3+n)-1/2*(224*n^2+896*n+845)/(3+n)/(2*n+3)*N+N^2, [[0, 147/4], [1, 173/4]], 1.071213028, 12.45227219, [6/7, 1/7, 2/7]] [`2F1(1,13/7;3;-1)`, .633204609994398485058078668818702885019665520591305296595\ 9697523455967262871106482955612651904005826, 1/49*(7*n+8)*(5+2*n)*(7*n+13)/(n+2 )/(2*n+3)/(3+n)-4/7*(21*n^2+84*n+80)/(3+n)/(2*n+3)*N+N^2, [[0, 3], [1, 34/7]], 1.634972864, 27.59157497, [6/7, 1/7, 1]] [`2F1(1,13/7;26/7;-1/7)`, .9342149953548615018301168370172001135820535695764087\ 890636847402584624967816492480654203274328937914, 7*(7*n+20)*(7*n+13)/(26+7*n)/ (n+2)-15*(7*n+20)*(14*n+33)/(26+7*n)/(n+2)*N+N^2, [[0, 182], [1, 195]], .576005\ 9371, 7.743583399, [6/7, 6/7, 1/7]] ----------------------- denominator , 8 [`2F1(1,1;2;-1/8)`, .9422642852510676363103528757641736405478457005178651288587\ 891103584617642250702954371322291056649196, 64*(1+n)/(n+2)-136*(2*n+3)/(n+2)*N+ N^2, [[0, 128], [1, 136]], -1.086717680, 2.783903493, [0, 0, 1/8]] 1571545212141185 Comment: This probably equals, arcsinh(----------------) 1444408272617472 [`2F1(1,1;2;-3/8)`, .8492099496494256421606592362415989215872172022844037676174\ 071484010473546777401766865726053690099434, 64/9*(1+n)/(n+2)-152/9*(2*n+3)/(n+2 )*N+N^2, [[0, 128/9], [1, 152/9]], 1.111115691, 11.49293718, [0, 0, 3/8]] 2/3 1/3 121 11 128 11 Comment: This probably equals, arcsinh(--------- - ---------) 512 1331 [`2F1(1,1;9/4;-1/8)`, .94837353137056893187361022283373204888949440934269856631\ 59694616828854909108614276950008260752220900, 64*(4*n+5)*(8*n+17)*(1+n)/(n+2)/( 8*n+9)/(9+4*n)-8*(8*n+13)*(136*n^2+442*n+325)/(n+2)/(8*n+9)/(9+4*n)*N+N^2, [[0, 144], [1, 152]], .5539025549, 7.903364178, [0, 1/4, 1/8]] [`2F1(1,1;5/2;-1/8)`, .95329850015803113902035637249435645071800483696918914834\ 44803417621703909090097618110797718710727515, 64*(2*n+3)*(4*n+9)*(1+n)/(n+2)/(4 *n+5)/(5+2*n)-8*(4*n+7)*(68*n^2+238*n+191)/(n+2)/(4*n+5)/(5+2*n)*N+N^2, [[0, 160], [1, 168]], -.9341686301e-1, 4.580141234, [0, 1/2, 1/8]] Comment: This probably equals, -24 + 36 ln(2) [`2F1(1,1;11/4;-1/8)`, .9573555694154233972999226272939264268076450127824589639\ 900704997991051805452251001858377930362119275, 64*(8*n+19)*(4*n+7)*(1+n)/(n+2)/ (8*n+11)/(11+4*n)-8*(8*n+15)*(136*n^2+510*n+443)/(n+2)/(8*n+11)/(11+4*n)*N+N^2, [[0, 176], [1, 184]], .5857922826, 8.196340681, [0, 3/4, 1/8]] [`2F1(1,9/8;9/4;-1/8)`, .942180507719058094234616699439282532574828535641212616\ 8461945453846652753459717536274814754668881925, 4*(8*n+17)*(8*n+9)/(9+4*n)/(n+2 )-17*(8*n+17)*(8*n+13)/(9+4*n)/(n+2)*N+N^2, [[0, 144], [1, 153]], .2802488205, 6.048176638, [1/8, 1/8, 1/8]] [`2F1(1,9/8;9/4;-1/4)`, .892289510371056573645308080625657318944180323512256562\ 8605403699703773251955443458409231232471483894, (8*n+17)*(8*n+9)/(9+4*n)/(n+2)-\ 9/2*(8*n+17)*(8*n+13)/(9+4*n)/(n+2)*N+N^2, [[0, 36], [1, 81/2]], .9705976592, 10.88757871, [1/8, 1/8, 1/4]] [`2F1(1,9/8;5/2;-1/8)`, .947680036064513350246640632533357225625692722260657722\ 6264940079180541433572667839743778486419776218, 2*(8*n+9)*(8*n+11)*(4*n+9)/(n+2 )/(4*n+5)/(5+2*n)-(4*n+7)*(544*n^2+1904*n+1529)/(n+2)/(4*n+5)/(5+2*n)*N+N^2, [[ 0, 160], [1, 169]], 1.178659537, 26.36935254, [1/8, 3/8, 1/8]] [`2F1(1,9/8;3;-1/8)`, .95601375887320661407346729859393632355633612773341023923\ 65425194195269389488798760195527572235673493, (8*n+9)*(5+2*n)*(8*n+15)/(n+2)/(2 *n+3)/(3+n)-2*(272*n^2+1088*n+1017)/(3+n)/(2*n+3)*N+N^2, [[0, 192], [1, 201]], .1434613264, 5.413046665, [1/8, 7/8, 1/8]] [`2F1(1,9/8;3;-1/4)`, .91703864097900562050032389605140483000781345014642301406\ 37273673811877489855033875595162592563585770, 1/4*(8*n+9)*(5+2*n)*(8*n+15)/(n+2 )/(2*n+3)/(3+n)-3*(48*n^2+192*n+179)/(3+n)/(2*n+3)*N+N^2, [[0, 48], [1, 105/2]] , .8357257105, 8.680247459, [1/8, 7/8, 1/4]] [`2F1(1,5/4;5/2;-1/8)`, .942108706813696845202411494924894292969394582140877743\ 2581195800402533193939234456305624572874903396, 8*(4*n+9)*(4*n+5)/(5+2*n)/(n+2) -34*(4*n+9)*(4*n+7)/(5+2*n)/(n+2)*N+N^2, [[0, 160], [1, 170]], -.3879126542, 3.\ 844668126, [1/4, 1/4, 1/8]] [`2F1(1,5/4;11/4;-1/8)`, .94710942125678791588025518822677178450700918604214086\ 50203448663421461405610702937168138264538210993, 32*(4*n+5)*(2*n+3)*(8*n+19)/(n +2)/(8*n+11)/(11+4*n)-2*(8*n+15)*(544*n^2+2040*n+1775)/(n+2)/(8*n+11)/(11+4*n)* N+N^2, [[0, 176], [1, 186]], 1.289784093, 45.12090915, [1/4, 1/2, 1/8]] [`2F1(1,5/4;3;-1/8)`, .95130532421789311087178284036141885110464501687690354762\ 01902283909619019511110386863382375424084884, 4*(4*n+7)*(5+2*n)*(4*n+5)/(n+2)/( 2*n+3)/(3+n)-4*(136*n^2+544*n+509)/(3+n)/(2*n+3)*N+N^2, [[0, 192], [1, 202]], -\ .5166971403, 3.592402881, [1/4, 3/4, 1/8]] [`2F1(1,11/8;5/2;-1/8)`, .93658405182949403293040973453910927361722812930113980\ 10884574386831825447409476591254895092493077597, 2*(8*n+9)*(8*n+11)*(4*n+9)/(n+ 2)/(4*n+5)/(5+2*n)-(4*n+7)*(544*n^2+1904*n+1531)/(n+2)/(4*n+5)/(5+2*n)*N+N^2, [ [0, 160], [1, 171]], 1.177389615, 26.24470943, [3/8, 1/8, 1/8]] [`2F1(1,11/8;11/4;-1/8)`, .9420464861852286642728528206117388999647136834060786\ 150198280464812425695748621818349854643016771462, 4*(8*n+19)*(8*n+11)/(11+4*n)/ (n+2)-17*(8*n+19)*(8*n+15)/(11+4*n)/(n+2)*N+N^2, [[0, 176], [1, 187]], .2983810\ 748, 6.143732775, [3/8, 3/8, 1/8]] [`2F1(1,11/8;11/4;-1/4)`, .8918342730299848987142927443675036880997629544959070\ 016273951206966894714397549024614206380613326634, (8*n+19)*(8*n+11)/(11+4*n)/(n +2)-9/2*(8*n+19)*(8*n+15)/(11+4*n)/(n+2)*N+N^2, [[0, 44], [1, 99/2]], .98793938\ 75, 11.25560163, [3/8, 3/8, 1/4]] [`2F1(1,11/8;3;-1/8)`, .9466315225131030958008632275595658602796147637976610925\ 184680824359054238344184272926064410889352341, (5+2*n)*(8*n+11)*(8*n+13)/(n+2)/ (2*n+3)/(3+n)-2*(272*n^2+1088*n+1019)/(3+n)/(2*n+3)*N+N^2, [[0, 192], [1, 203]] , .1383047862, 5.391702653, [3/8, 5/8, 1/8]] [`2F1(1,11/8;3;-1/4)`, .8999943417957442119219970611098413845028973903375130330\ 732592583019147503834196035524311908499710708, 1/4*(5+2*n)*(8*n+11)*(8*n+13)/(n +2)/(2*n+3)/(3+n)-(144*n^2+576*n+539)/(3+n)/(2*n+3)*N+N^2, [[0, 48], [1, 107/2] ], .8347689320, 8.667781262, [3/8, 5/8, 1/4]] [`2F1(1,3/2;5/2;-1/8)`, .931105615441117029080165279406114891260881748235253023\ 7539077363674580470563973635668956092613233270, 64*(2*n+3)*(4*n+9)*(1+n)/(n+2)/ (4*n+5)/(5+2*n)-4*(4*n+7)*(136*n^2+476*n+383)/(n+2)/(4*n+5)/(5+2*n)*N+N^2, [[0, 160], [1, 172]], 1.292098483, 45.79920999, [1/2, 0, 1/8]] [`2F1(1,3/2;21/8;-1/8)`, .93419936935584501678029649746902716527767813504570599\ 82746012678536135135448803477791128421989870789, 32*(2*n+3)*(8*n+9)*(16*n+37)/( n+2)/(16*n+21)/(21+8*n)-4*(16*n+29)*(544*n^2+1972*n+1653)/(n+2)/(16*n+21)/(21+8 *n)*N+N^2, [[0, 168], [1, 180]], 1.301838959, 48.89257413, [1/2, 1/8, 1/8]] [`2F1(1,3/2;11/4;-1/8)`, .93702314905812305928855071483291999866919986657061777\ 80706036631381174437997983801784200498085699310, 32*(4*n+5)*(2*n+3)*(8*n+19)/(n +2)/(8*n+11)/(11+4*n)-4*(8*n+15)*(272*n^2+1020*n+889)/(n+2)/(8*n+11)/(11+4*n)*N +N^2, [[0, 176], [1, 188]], .5988244518, 8.322417775, [1/2, 1/4, 1/8]] [`2F1(1,3/2;23/8;-1/8)`, .93961116671163483837804149571451559029051339966896791\ 27700049544513723253712927466459824693344143072, 32*(8*n+11)*(16*n+39)*(2*n+3)/ (n+2)/(16*n+23)/(23+8*n)-4*(16*n+31)*(544*n^2+2108*n+1907)/(n+2)/(16*n+23)/(23+ 8*n)*N+N^2, [[0, 184], [1, 196]], 1.291617230, 45.65649027, [1/2, 3/8, 1/8]] [`2F1(1,3/2;3;-1/8)`, .94199204873150126015152990347593782924599985525193990015\ 49806115587282705508970814511868181856360698, 16*(5+2*n)*(2*n+3)/(3+n)/(n+2)-\ 136*(5+2*n)/(3+n)*N+N^2, [[0, 192], [1, 204]], -2.078147338, 2.000734452, [1/2, 1/2, 1/8]] [`2F1(1,3/2;3;-3/8)`, .84742030473344633506659394070092400941186341814418270427\ 44619944245385563983684055876893004580531861, 16/9*(5+2*n)*(2*n+3)/(3+n)/(n+2)-\ 152/9*(5+2*n)/(3+n)*N+N^2, [[0, 64/3], [1, 76/3]], .1190772389, 3.535350704, [1 /2, 1/2, 3/8]] 1/2 128 22 Comment: This probably equals, 608/9 - --------- 9 [`2F1(1,3/2;3;-5/8)`, .77302009040494167025549256180335621343349275597006639316\ 49494813392198038414121185876538823990681172, 16/25*(5+2*n)*(2*n+3)/(3+n)/(n+2) -168/25*(5+2*n)/(3+n)*N+N^2, [[0, 192/25], [1, 252/25]], 1.140728486, 8.4042961\ 44, [1/2, 1/2, 5/8]] 1/2 672 128 26 Comment: This probably equals, --- - --------- 25 25 [`2F1(1,3/2;25/8;-1/8)`, .94418993643096221312873297344220872252832333724761521\ 08789525040512882893290148681839246840594848384, 32*(2*n+3)*(16*n+41)*(8*n+13)/ (n+2)/(16*n+25)/(25+8*n)-4*(16*n+33)*(544*n^2+2244*n+2177)/(n+2)/(16*n+25)/(25+ 8*n)*N+N^2, [[0, 200], [1, 212]], 1.256543282, 37.20650374, [1/2, 5/8, 1/8]] [`2F1(1,3/2;13/4;-1/8)`, .94622533541973463573062094307783634891096411876619915\ 64669544751623062574524169813948004751783132938, 32*(2*n+3)*(4*n+7)*(8*n+21)/(n +2)/(8*n+13)/(13+4*n)-4*(8*n+17)*(272*n^2+1156*n+1159)/(n+2)/(8*n+13)/(13+4*n)* N+N^2, [[0, 208], [1, 220]], .6075539052, 8.409060885, [1/2, 3/4, 1/8]] [`2F1(1,3/2;27/8;-1/8)`, .94811577837136287644447046674737275236471355172227097\ 60275365332807065838487170556739061692522784694, 32*(8*n+15)*(2*n+3)*(16*n+43)/ (n+2)/(16*n+27)/(27+8*n)-4*(16*n+35)*(544*n^2+2380*n+2463)/(n+2)/(16*n+27)/(27+ 8*n)*N+N^2, [[0, 216], [1, 228]], 1.264025101, 38.73578878, [1/2, 7/8, 1/8]] [`2F1(1,13/8;3;-1/8)`, .9373866008270484851005178353954069677280654320656336306\ 223172783808436960179540841859470609741920293, (5+2*n)*(8*n+11)*(8*n+13)/(n+2)/ (2*n+3)/(3+n)-2*(272*n^2+1088*n+1021)/(3+n)/(2*n+3)*N+N^2, [[0, 192], [1, 205]] , .1419879556, 5.406930849, [5/8, 3/8, 1/8]] [`2F1(1,13/8;3;-1/4)`, .8834200889356395677765785255254021205417423431785752276\ 572210232664656951114739889555966237469145121, 1/4*(5+2*n)*(8*n+11)*(8*n+13)/(n +2)/(2*n+3)/(3+n)-(144*n^2+576*n+541)/(3+n)/(2*n+3)*N+N^2, [[0, 48], [1, 109/2] ], .8300114167, 8.606321880, [5/8, 3/8, 1/4]] [`2F1(1,13/8;13/4;-1/8)`, .9419440203185083776328280142725484616559796162342952\ 516701217081455362080630337938667840505129303889, 4*(8*n+21)*(8*n+13)/(13+4*n)/ (n+2)-17*(8*n+21)*(8*n+17)/(13+4*n)/(n+2)*N+N^2, [[0, 208], [1, 221]], .2802881\ 334, 6.048380600, [5/8, 5/8, 1/8]] [`2F1(1,13/8;13/4;-1/4)`, .8914864085011093703680148579091398617898614330587538\ 174946650847487947556348172693559042481190474261, (8*n+21)*(8*n+13)/(13+4*n)/(n +2)-9/2*(8*n+21)*(8*n+17)/(13+4*n)/(n+2)*N+N^2, [[0, 52], [1, 117/2]], .9759154\ 411, 10.99784734, [5/8, 5/8, 1/4]] [`2F1(1,13/8;7/2;-1/8)`, .94587577557327938512778965834831037024417765210196013\ 59043952738735682187959492231907844026239876766, 2*(8*n+15)*(4*n+11)*(8*n+13)/( n+2)/(4*n+7)/(7+2*n)-(4*n+9)*(544*n^2+2448*n+2615)/(n+2)/(4*n+7)/(7+2*n)*N+N^2, [[0, 224], [1, 237]], 1.179121953, 26.41503331, [5/8, 7/8, 1/8]] [`2F1(1,7/4;3;-1/8)`, .93281487970380600993203022472165372494754332828874117215\ 52395525956095918933823566776144652381776425, 4*(4*n+7)*(5+2*n)*(4*n+5)/(n+2)/( 2*n+3)/(3+n)-4*(136*n^2+544*n+511)/(3+n)/(2*n+3)*N+N^2, [[0, 192], [1, 206]], -\ .5142017894, 3.596975913, [3/4, 1/4, 1/8]] [`2F1(1,7/4;13/4;-1/8)`, .93769226321344939074550596612399733180981798565209352\ 14468226200935523591971596297967951003639421589, 32*(2*n+3)*(4*n+7)*(8*n+21)/(n +2)/(8*n+13)/(13+4*n)-2*(8*n+17)*(544*n^2+2312*n+2323)/(n+2)/(8*n+13)/(13+4*n)* N+N^2, [[0, 208], [1, 222]], 1.298918964, 47.92226017, [3/4, 1/2, 1/8]] [`2F1(1,7/4;7/2;-1/8)`, .941901332267327761773044304421286938428457526411178114\ 0265346811449490982398069327710859311145738387, 8*(4*n+11)*(4*n+7)/(7+2*n)/(n+2 )-34*(4*n+11)*(4*n+9)/(7+2*n)/(n+2)*N+N^2, [[0, 224], [1, 238]], -.4011632003, 3.817089354, [3/4, 3/4, 1/8]] [`2F1(1,15/8;3;-1/8)`, .9282765891850821720581261582694098721173197705403622533\ 790987870920586620442745298295312250321618952, (8*n+9)*(5+2*n)*(8*n+15)/(n+2)/( 2*n+3)/(3+n)-2*(272*n^2+1088*n+1023)/(3+n)/(2*n+3)*N+N^2, [[0, 192], [1, 207]], .1413627535, 5.404339870, [7/8, 1/8, 1/8]] [`2F1(1,15/8;3;-1/4)`, .8673004111317121139054844231130537356546663942730568591\ 006970365388292863228361130861594047607878222, 1/4*(8*n+9)*(5+2*n)*(8*n+15)/(n+ 2)/(2*n+3)/(3+n)-3*(48*n^2+192*n+181)/(3+n)/(2*n+3)*N+N^2, [[0, 48], [1, 111/2] ], .8385469043, 8.717215397, [7/8, 1/8, 1/4]] [`2F1(1,15/8;7/2;-1/8)`, .93795280842240578955047713242319515059748951605052511\ 34857615044219935336829203915146188941810306867, 2*(8*n+15)*(4*n+11)*(8*n+13)/( n+2)/(4*n+7)/(7+2*n)-(4*n+9)*(544*n^2+2448*n+2621)/(n+2)/(4*n+7)/(7+2*n)*N+N^2, [[0, 224], [1, 239]], 1.174116693, 25.92883792, [7/8, 5/8, 1/8]] [`2F1(1,15/8;15/4;-1/8)`, .9418631409550614056576763792492947377417289358341627\ 050709192449742184897759735615915051599501112401, 4*(8*n+23)*(8*n+15)/(15+4*n)/ (n+2)-17*(8*n+23)*(8*n+19)/(15+4*n)/(n+2)*N+N^2, [[0, 240], [1, 255]], .2881243\ 413, 6.089312635, [7/8, 7/8, 1/8]] [`2F1(1,15/8;15/4;-1/4)`, .8912119581052994929153615290818744446319002940913483\ 596695380911146947133045224535471108913519279578, (8*n+23)*(8*n+15)/(15+4*n)/(n +2)-9/2*(8*n+23)*(8*n+19)/(15+4*n)/(n+2)*N+N^2, [[0, 60], [1, 135/2]], .9829900\ 556, 11.14805499, [7/8, 7/8, 1/4]] ----------------------- denominator , 9 [`2F1(1,1;2;-1/9)`, .9482446409204367110475088275538151847550833568494666530754\ 531032332562091612109183988912591592070869, 81*(1+n)/(n+2)-171*(2*n+3)/(n+2)*N+ N^2, [[0, 162], [1, 171]], -1.208915704, 2.746292387, [0, 0, 1/9]] 849905364703000879 Comment: This probably equals, arcsinh(------------------) 774840978000000000 [`2F1(1,1;2;-2/9)`, .9030181295796802257215389685403505073702658223136322228713\ 056883501326686420845069752043926882677623, 81/4*(1+n)/(n+2)-45*(2*n+3)/(n+2)*N +N^2, [[0, 81/2], [1, 45]], .1761488088, 4.559376363, [0, 0, 2/9]] 985263601 Comment: This probably equals, arctanh(----------) 1372684090 [`2F1(1,1;2;-4/9)`, .8273807552819640448417431773706011898981453655080934791367\ 278976240711819961825527177720960793069502, 81/16*(1+n)/(n+2)-99/8*(2*n+3)/(n+2 )*N+N^2, [[0, 81/8], [1, 99/8]], 1.566258538, 387.3549458, [0, 0, 4/9]] 1/2 1/2 (-570036489282 + 428684874618 13 ) Comment: This probably equals, arcsinh(---------------------------------------) 1067742 [`2F1(1,1;7/3;-1/9)`, .95533006869220400917806216866143263975195396486787866512\ 64842475697958742778189743297713458072502555, 81*(3*n+4)*(6*n+13)*(1+n)/(n+2)/( 6*n+7)/(7+3*n)-36*(3*n+5)*(57*n^2+190*n+144)/(n+2)/(6*n+7)/(7+3*n)*N+N^2, [[0, 189], [1, 198]], .5078269115e-1, 5.237138647, [0, 1/3, 1/9]] [`2F1(1,1;7/3;-2/9)`, .91578394973301858819565774690272139917255651191620949453\ 14447341979094190617516792465528297457938962, 81/4*(3*n+4)*(6*n+13)*(1+n)/(n+2) /(6*n+7)/(7+3*n)-9/2*(3*n+5)*(120*n^2+400*n+303)/(n+2)/(6*n+7)/(7+3*n)*N+N^2, [ [0, 189/4], [1, 207/4]], 1.436280873, 113.2412472, [0, 1/3, 2/9]] [`2F1(1,1;8/3;-1/9)`, .96069127291695070870111745024352147717831065007187355615\ 25689591465000430303588686361785957454760804, 81*(3*n+5)*(3*n+7)*(1+n)/(n+2)/(3 *n+4)/(8+3*n)-9*(6*n+11)*(57*n^2+209*n+177)/(n+2)/(3*n+4)/(8+3*n)*N+N^2, [[0, 216], [1, 225]], .3900672340e-1, 5.193107751, [0, 2/3, 1/9]] [`2F1(1,1;8/3;-2/9)`, .92551886876710922260778004751048408430870680272711565887\ 58363782121113887249238751640885564662695122, 81/4*(3*n+5)*(3*n+7)*(1+n)/(n+2)/ (3*n+4)/(8+3*n)-9/2*(6*n+11)*(30*n^2+110*n+93)/(n+2)/(3*n+4)/(8+3*n)*N+N^2, [[0 , 54], [1, 117/2]], 1.422170073, 89.38284259, [0, 2/3, 2/9]] [`2F1(1,10/9;20/9;-1/9)`, .9481841541692998144031708631760172146889697328032824\ 543652461328703965764252007666303188760305608774, 9*(9*n+19)*(9*n+10)/(20+9*n)/ (n+2)-19*(9*n+19)*(18*n+29)/(20+9*n)/(n+2)*N+N^2, [[0, 180], [1, 190]], .812578\ 3723, 11.59927346, [1/9, 1/9, 1/9]] [`2F1(1,10/9;7/3;-1/9)`, .95053958242270752473538745325899153271924753633563670\ 59052947679820706381097631366159023312835923309, 3*(9*n+10)*(9*n+11)*(6*n+13)/( n+2)/(6*n+7)/(7+3*n)-2*(3*n+5)*(1026*n^2+3420*n+2593)/(n+2)/(6*n+7)/(7+3*n)*N+N ^2, [[0, 189], [1, 199]], 1.170084706, 26.98157555, [1/9, 2/9, 1/9]] [`2F1(1,10/9;8/3;-1/9)`, .95646233382124738021437302846916224171900348322930828\ 97813829624987036295095476736959929567089168966, 3*(9*n+10)*(3*n+7)*(9*n+14)/(n +2)/(3*n+4)/(8+3*n)-(6*n+11)*(513*n^2+1881*n+1594)/(n+2)/(3*n+4)/(8+3*n)*N+N^2, [[0, 216], [1, 226]], 1.158728138, 25.89088678, [1/9, 5/9, 1/9]] [`2F1(1,10/9;3;-1/9)`, .9611058965488831608902863276405443641703953396209909394\ 346483382465208324384342026952647228541597502, (9*n+17)*(5+2*n)*(9*n+10)/(n+2)/ (2*n+3)/(3+n)-2*(342*n^2+1368*n+1279)/(3+n)/(2*n+3)*N+N^2, [[0, 243], [1, 253]] , -.4178256863e-1, 4.909907306, [1/9, 8/9, 1/9]] [`2F1(1,10/9;3;-2/9)`, .9261887576169442125409313935009700082244849780817754138\ 576422188867914999343072320054568743772124475, 1/4*(9*n+17)*(5+2*n)*(9*n+10)/(n +2)/(2*n+3)/(3+n)-1/2*(360*n^2+1440*n+1343)/(3+n)/(2*n+3)*N+N^2, [[0, 243/4], [ 1, 263/4]], 1.343662695, 41.14884586, [1/9, 8/9, 2/9]] [`2F1(1,10/9;3;-1/3)`, .8945867049882922610962213440754638638366413387405577481\ 440420762571211380207726968521926513184699905, 1/9*(9*n+17)*(5+2*n)*(9*n+10)/(n +2)/(2*n+3)/(3+n)-14/3*(18*n^2+72*n+67)/(3+n)/(2*n+3)*N+N^2, [[0, 27], [1, 91/3 ]], 1.054420599, 10.95449868, [1/9, 8/9, 1/3]] [`2F1(1,11/9;7/3;-1/9)`, .94578264444756621449742593144370534838070741816168216\ 74753071936683055412223257395207014676159148844, 3*(9*n+10)*(9*n+11)*(6*n+13)/( n+2)/(6*n+7)/(7+3*n)-4*(3*n+5)*(513*n^2+1710*n+1297)/(n+2)/(6*n+7)/(7+3*n)*N+N^ 2, [[0, 189], [1, 200]], 1.168591410, 26.83294059, [2/9, 1/9, 1/9]] [`2F1(1,11/9;22/9;-1/9)`, .9481314765780479594945197641604320767217299054646572\ 363466735996704340962012899160200925637368999519, 9*(9*n+20)*(9*n+11)/(22+9*n)/ (n+2)-19*(9*n+20)*(18*n+31)/(22+9*n)/(n+2)*N+N^2, [[0, 198], [1, 209]], .820225\ 6747, 11.74247183, [2/9, 2/9, 1/9]] [`2F1(1,11/9;8/3;-1/9)`, .95226037294895365187200407196247802087676093174368527\ 06480198923989406913975047339029773794183124438, 3*(9*n+11)*(9*n+13)*(3*n+7)/(n +2)/(3*n+4)/(8+3*n)-(6*n+11)*(513*n^2+1881*n+1595)/(n+2)/(3*n+4)/(8+3*n)*N+N^2, [[0, 216], [1, 227]], 1.147786035, 24.92028684, [2/9, 4/9, 1/9]] [`2F1(1,11/9;3;-1/9)`, .9573411654528138964306305322073403782219555997742832894\ 046127189563875435012872264760220269254671731, (9*n+11)*(5+2*n)*(9*n+16)/(n+2)/ (2*n+3)/(3+n)-4*(171*n^2+684*n+640)/(3+n)/(2*n+3)*N+N^2, [[0, 243], [1, 254]], -.4207176305e-1, 4.908949030, [2/9, 7/9, 1/9]] [`2F1(1,11/9;3;-2/9)`, .9192564037095387780942300511289282660175081160102356179\ 949656961190284505584209181025748543541588133, 1/4*(9*n+11)*(5+2*n)*(9*n+16)/(n +2)/(2*n+3)/(3+n)-5/2*(72*n^2+288*n+269)/(3+n)/(2*n+3)*N+N^2, [[0, 243/4], [1, 265/4]], 1.343511563, 41.10614351, [2/9, 7/9, 2/9]] [`2F1(1,11/9;3;-1/3)`, .8849604532256937884023569931198404600399217500183889482\ 717530161775999858847297410587209388962051519, 1/9*(9*n+11)*(5+2*n)*(9*n+16)/(n +2)/(2*n+3)/(3+n)-4/3*(63*n^2+252*n+235)/(3+n)/(2*n+3)*N+N^2, [[0, 27], [1, 92/ 3]], 1.059513645, 11.07176003, [2/9, 7/9, 1/3]] [`2F1(1,4/3;22/9;-1/9)`, .94360633385849853807826878763286854939006301694226507\ 62948836112433894544073732586981982716910761475, 27*(9*n+10)*(3*n+4)*(9*n+20)/( n+2)/(9*n+11)/(22+9*n)-3*(18*n+31)*(513*n^2+1767*n+1394)/(n+2)/(9*n+11)/(22+9*n )*N+N^2, [[0, 198], [1, 210]], 1.325295575, 63.59729725, [1/3, 1/9, 1/9]] [`2F1(1,4/3;23/9;-1/9)`, .94593969653898567121095957050284939432758791636389889\ 43736030604734649510148901916017337183584379936, 27*(9*n+11)*(3*n+4)*(18*n+41)/ (n+2)/(18*n+23)/(23+9*n)-6*(9*n+16)*(1026*n^2+3648*n+2987)/(n+2)/(18*n+23)/(23+ 9*n)*N+N^2, [[0, 207], [1, 219]], 1.402755151, 197.0522304, [1/3, 2/9, 1/9]] [`2F1(1,4/3;8/3;-1/9)`, .948085187810853555024835567180335395371006279862670725\ 3539135306138519677717590981258146419093480561, 27*(3*n+7)*(3*n+4)/(8+3*n)/(n+2 )-57*(3*n+7)*(6*n+11)/(8+3*n)/(n+2)*N+N^2, [[0, 216], [1, 228]], -.3204347887, 4.132591075, [1/3, 1/3, 1/9]] [`2F1(1,4/3;8/3;-2/9)`, .902467625473148044958728527580272455837326000205070515\ 6151539646506285751942664441928293515493454695, 27/4*(3*n+7)*(3*n+4)/(8+3*n)/(n +2)-15*(3*n+7)*(6*n+11)/(8+3*n)/(n+2)*N+N^2, [[0, 54], [1, 60]], 1.066368371, 14.15981748, [1/3, 1/3, 2/9]] [`2F1(1,4/3;25/9;-1/9)`, .95006481449554319061380343767777527383719812975529029\ 45625896032902715937401797173989552881624788570, 27*(3*n+4)*(9*n+13)*(18*n+43)/ (n+2)/(18*n+25)/(25+9*n)-6*(9*n+17)*(1026*n^2+3876*n+3403)/(n+2)/(18*n+25)/(25+ 9*n)*N+N^2, [[0, 225], [1, 237]], 1.304871389, 53.96113704, [1/3, 4/9, 1/9]] [`2F1(1,4/3;26/9;-1/9)`, .95189725526656420938500532068091461320987524899913878\ 26445644391310239451201455433930630664202194993, 27*(9*n+22)*(3*n+4)*(9*n+14)/( n+2)/(9*n+13)/(26+9*n)-3*(18*n+35)*(513*n^2+1995*n+1810)/(n+2)/(9*n+13)/(26+9*n )*N+N^2, [[0, 234], [1, 246]], 1.410200677, 246.8413080, [1/3, 5/9, 1/9]] [`2F1(1,4/3;3;-1/9)`, .95359846943989613345444460759579673364051924730667860866\ 99988771400724744308436918298658984825754045, 9*(3*n+5)*(5+2*n)*(3*n+4)/(n+2)/( 2*n+3)/(3+n)-6*(114*n^2+456*n+427)/(3+n)/(2*n+3)*N+N^2, [[0, 243], [1, 255]], -\ 1.297879417, 2.657044488, [1/3, 2/3, 1/9]] (2/3) Comment: This probably equals, -81 30 + 783 [`2F1(1,4/3;3;-2/9)`, .91240131843904549530523859030913797936713011779653629402\ 79670265112206317672288879469089950627693958, 9/4*(3*n+5)*(5+2*n)*(3*n+4)/(n+2) /(2*n+3)/(3+n)-3/2*(120*n^2+480*n+449)/(3+n)/(2*n+3)*N+N^2, [[0, 243/4], [1, 267/4]], .9096468324e-1, 4.281596471, [1/3, 2/3, 2/9]] [`2F1(1,4/3;28/9;-1/9)`, .95518217646824766405016572470764077133965970204685502\ 39355490116651686834053532486317617254825252232, 27*(9*n+16)*(3*n+4)*(9*n+23)/( n+2)/(9*n+14)/(28+9*n)-3*(18*n+37)*(513*n^2+2109*n+2036)/(n+2)/(9*n+14)/(28+9*n )*N+N^2, [[0, 252], [1, 264]], 1.334078216, 68.88712262, [1/3, 7/9, 1/9]] [`2F1(1,4/3;29/9;-1/9)`, .95666023714507489061020602924660995717818599792333321\ 12472851879241188979827951593305624210341200395, 27*(9*n+17)*(3*n+4)*(18*n+47)/ (n+2)/(18*n+29)/(29+9*n)-6*(9*n+19)*(1026*n^2+4332*n+4307)/(n+2)/(18*n+29)/(29+ 9*n)*N+N^2, [[0, 261], [1, 273]], 1.400477367, 185.5994712, [1/3, 8/9, 1/9]] [`2F1(1,13/9;8/3;-1/9)`, .94393657758496272757819252671856655233256871462447558\ 79017922435983548368529029116767092278189985536, 3*(9*n+11)*(9*n+13)*(3*n+7)/(n +2)/(3*n+4)/(8+3*n)-(6*n+11)*(513*n^2+1881*n+1597)/(n+2)/(3*n+4)/(8+3*n)*N+N^2, [[0, 216], [1, 229]], 1.146355218, 24.79872257, [4/9, 2/9, 1/9]] [`2F1(1,13/9;26/9;-1/9)`, .9480441922496454712385977845492685125533223821754656\ 641152255790470349077337042684757505831783703826, 9*(9*n+22)*(9*n+13)/(26+9*n)/ (n+2)-19*(9*n+22)*(18*n+35)/(26+9*n)/(n+2)*N+N^2, [[0, 234], [1, 247]], .813926\ 2651, 11.62425913, [4/9, 4/9, 1/9]] [`2F1(1,13/9;3;-1/9)`, .9498776540414368628946913628262230259012680884558606853\ 547753203253512062605357668177383872378802415, (5+2*n)*(9*n+13)*(9*n+14)/(n+2)/ (2*n+3)/(3+n)-4*(171*n^2+684*n+641)/(3+n)/(2*n+3)*N+N^2, [[0, 243], [1, 256]], -.4886285684e-1, 4.886553034, [4/9, 5/9, 1/9]] [`2F1(1,13/9;3;-2/9)`, .9056224720516206214537009680241770508285607040115172446\ 112898589903172976956640138995278633805383639, 1/4*(5+2*n)*(9*n+13)*(9*n+14)/(n +2)/(2*n+3)/(3+n)-1/2*(360*n^2+1440*n+1349)/(3+n)/(2*n+3)*N+N^2, [[0, 243/4], [ 1, 269/4]], 1.337214584, 39.40244363, [4/9, 5/9, 2/9]] [`2F1(1,13/9;3;-1/3)`, .8661663734230163298081954940172876049217639470490803654\ 007370934654946365003346996873861760191435168, 1/9*(5+2*n)*(9*n+13)*(9*n+14)/(n +2)/(2*n+3)/(3+n)-4/3*(63*n^2+252*n+236)/(3+n)/(2*n+3)*N+N^2, [[0, 27], [1, 94/ 3]], 1.049174190, 10.83627546, [4/9, 5/9, 1/3]] [`2F1(1,13/9;10/3;-1/9)`, .9546687296786766515432894960402366532310240647008288\ 492571511560758263835748401491555142054029024853, 3*(9*n+17)*(9*n+13)*(3*n+8)/( n+2)/(3*n+5)/(10+3*n)-(6*n+13)*(513*n^2+2223*n+2276)/(n+2)/(3*n+5)/(10+3*n)*N+N ^2, [[0, 270], [1, 283]], 1.145510155, 24.72748030, [4/9, 8/9, 1/9]] [`2F1(1,14/9;8/3;-1/9)`, .93981434310195143029325048169170968101094927330102519\ 61026669988817105309782711515539654743339899472, 3*(9*n+10)*(3*n+7)*(9*n+14)/(n +2)/(3*n+4)/(8+3*n)-(6*n+11)*(513*n^2+1881*n+1598)/(n+2)/(3*n+4)/(8+3*n)*N+N^2, [[0, 216], [1, 230]], 1.162519497, 26.24507091, [5/9, 1/9, 1/9]] [`2F1(1,14/9;3;-1/9)`, .9461785659905801651476992149615805778256152627757954364\ 817286553685473981941078270976820356803708360, (5+2*n)*(9*n+13)*(9*n+14)/(n+2)/ (2*n+3)/(3+n)-2*(342*n^2+1368*n+1283)/(3+n)/(2*n+3)*N+N^2, [[0, 243], [1, 257]] , -.4965237119e-1, 4.883962589, [5/9, 4/9, 1/9]] [`2F1(1,14/9;3;-2/9)`, .8989188500101296818106779091292606095588346906776110846\ 770145154472305023142666582954501103569297682, 1/4*(5+2*n)*(9*n+13)*(9*n+14)/(n +2)/(2*n+3)/(3+n)-1/2*(360*n^2+1440*n+1351)/(3+n)/(2*n+3)*N+N^2, [[0, 243/4], [ 1, 271/4]], 1.336955535, 39.33537502, [5/9, 4/9, 2/9]] [`2F1(1,14/9;3;-1/3)`, .8569927411942547505185295153080407467760543549706838334\ 202153223447925354222647882990426186117823765, 1/9*(5+2*n)*(9*n+13)*(9*n+14)/(n +2)/(2*n+3)/(3+n)-2/3*(126*n^2+504*n+473)/(3+n)/(2*n+3)*N+N^2, [[0, 27], [1, 95 /3]], 1.050527671, 10.86653004, [5/9, 4/9, 1/3]] [`2F1(1,14/9;28/9;-1/9)`, .9480076312205840820729464451216830328697347512388977\ 005268815595065245206618288449100664632384884524, 9*(9*n+23)*(9*n+14)/(28+9*n)/ (n+2)-19*(9*n+23)*(18*n+37)/(28+9*n)/(n+2)*N+N^2, [[0, 252], [1, 266]], .833436\ 0002, 11.99835051, [5/9, 5/9, 1/9]] [`2F1(1,14/9;10/3;-1/9)`, .9513127239068820563853807676031147586587480261541756\ 688180955400630013410713474286896358592244870717, 3*(3*n+8)*(9*n+14)*(9*n+16)/( n+2)/(3*n+5)/(10+3*n)-(6*n+13)*(513*n^2+2223*n+2278)/(n+2)/(3*n+5)/(10+3*n)*N+N ^2, [[0, 270], [1, 284]], 1.146141738, 24.78068661, [5/9, 7/9, 1/9]] [`2F1(1,5/3;25/9;-1/9)`, .93815284740117915491209132484888012974205669126994769\ 59420608215304811099668225433953555498712046368, 27*(3*n+5)*(9*n+10)*(18*n+43)/ (n+2)/(18*n+25)/(25+9*n)-12*(9*n+17)*(513*n^2+1938*n+1705)/(n+2)/(18*n+25)/(25+ 9*n)*N+N^2, [[0, 225], [1, 240]], 1.377844471, 117.6537474, [2/3, 1/9, 1/9]] [`2F1(1,5/3;26/9;-1/9)`, .94040728219471965676569949640285462126595954868100602\ 77940516381035566572064690799938331645068271627, 27*(3*n+5)*(9*n+11)*(9*n+22)/( n+2)/(9*n+13)/(26+9*n)-3*(18*n+35)*(513*n^2+1995*n+1814)/(n+2)/(9*n+13)/(26+9*n )*N+N^2, [[0, 234], [1, 249]], 1.340071733, 73.03261523, [2/3, 2/9, 1/9]] [`2F1(1,5/3;3;-1/9)`, .94250105321229534872803011206011382625764053661645364253\ 76677650312343881519744234108932634194408384, 9*(3*n+5)*(5+2*n)*(3*n+4)/(n+2)/( 2*n+3)/(3+n)-12*(57*n^2+228*n+214)/(3+n)/(2*n+3)*N+N^2, [[0, 243], [1, 258]], -\ 1.295329678, 2.659521553, [2/3, 1/3, 1/9]] [`2F1(1,5/3;3;-2/9)`, .89228945275354088118470331803915323476751145084524056776\ 26352775082399056843561661990611447395186919, 9/4*(3*n+5)*(5+2*n)*(3*n+4)/(n+2) /(2*n+3)/(3+n)-3/2*(120*n^2+480*n+451)/(3+n)/(2*n+3)*N+N^2, [[0, 243/4], [1, 273/4]], .9002027879e-1, 4.278706410, [2/3, 1/3, 2/9]] 1/3 243 33 Comment: This probably equals, - --------- + 783/4 4 [`2F1(1,5/3;3;-4/9)`, .80847620922234230281343834895706211357067635444926452808\ 31970443221307708461063158448929989418467101, 9/16*(3*n+5)*(5+2*n)*(3*n+4)/(n+2 )/(2*n+3)/(3+n)-3/8*(132*n^2+528*n+497)/(3+n)/(2*n+3)*N+N^2, [[0, 243/16], [1, 303/16]], 1.477259044, 47.15992854, [2/3, 1/3, 4/9]] 1/3 243 39 837 Comment: This probably equals, - --------- + --- 16 16 [`2F1(1,5/3;28/9;-1/9)`, .94445089801138995199007109850388434338775466720247210\ 88843319337261806115889891269928098997647590529, 27*(9*n+13)*(3*n+5)*(9*n+23)/( n+2)/(9*n+14)/(28+9*n)-3*(18*n+37)*(513*n^2+2109*n+2041)/(n+2)/(9*n+14)/(28+9*n )*N+N^2, [[0, 252], [1, 267]], 1.398019648, 174.6470599, [2/3, 4/9, 1/9]] [`2F1(1,5/3;29/9;-1/9)`, .94627129250804788340294848588687063651631087837450360\ 41288556692419554629308049557531622485370616469, 27*(3*n+5)*(18*n+47)*(9*n+14)/ (n+2)/(18*n+29)/(29+9*n)-12*(9*n+19)*(513*n^2+2166*n+2159)/(n+2)/(18*n+29)/(29+ 9*n)*N+N^2, [[0, 261], [1, 276]], 1.321930895, 61.77982168, [2/3, 5/9, 1/9]] [`2F1(1,5/3;10/3;-1/9)`, .94797482223016598193201667483609059462429210308250029\ 28736279692009289417823818609175553728328949747, 27*(3*n+8)*(3*n+5)/(10+3*n)/(n +2)-57*(3*n+8)*(6*n+13)/(10+3*n)/(n+2)*N+N^2, [[0, 270], [1, 285]], -.347881943\ 2, 4.069136704, [2/3, 2/3, 1/9]] [`2F1(1,5/3;10/3;-2/9)`, .90208682056475325968032138992919279401602429623410759\ 10444987778750816152070659752362963647860984095, 27/4*(3*n+8)*(3*n+5)/(10+3*n)/ (n+2)-15*(3*n+8)*(6*n+13)/(10+3*n)/(n+2)*N+N^2, [[0, 135/2], [1, 75]], 1.039341\ 651, 13.30901662, [2/3, 2/3, 2/9]] [`2F1(1,5/3;32/9;-1/9)`, .95107392313348291137186242346913601929691146399743046\ 14305162900576189290666517680743289383576889918, 27*(3*n+5)*(9*n+17)*(9*n+25)/( n+2)/(9*n+16)/(32+9*n)-3*(18*n+41)*(513*n^2+2337*n+2531)/(n+2)/(9*n+16)/(32+9*n )*N+N^2, [[0, 288], [1, 303]], 1.303638053, 53.47189075, [2/3, 8/9, 1/9]] [`2F1(1,16/9;3;-1/9)`, .9388449648134526223079811348797885740509371617849511556\ 682326628547656695997027310347553494731130946, (9*n+11)*(5+2*n)*(9*n+16)/(n+2)/ (2*n+3)/(3+n)-2*(342*n^2+1368*n+1285)/(3+n)/(2*n+3)*N+N^2, [[0, 243], [1, 259]] , -.4708179634e-1, 4.892406909, [7/9, 2/9, 1/9]] [`2F1(1,16/9;3;-2/9)`, .8857332954603097476396214499346009484613614834474213197\ 153869653126580759441761314538682802808840175, 1/4*(9*n+11)*(5+2*n)*(9*n+16)/(n +2)/(2*n+3)/(3+n)-5/2*(72*n^2+288*n+271)/(3+n)/(2*n+3)*N+N^2, [[0, 243/4], [1, 275/4]], 1.344939334, 41.51313163, [7/9, 2/9, 2/9]] [`2F1(1,16/9;3;-1/3)`, .8390782380491638299780760348643327583849106365531572630\ 512374447961081922191983046672817310670230620, 1/9*(9*n+11)*(5+2*n)*(9*n+16)/(n +2)/(2*n+3)/(3+n)-2/3*(126*n^2+504*n+475)/(3+n)/(2*n+3)*N+N^2, [[0, 27], [1, 97 /3]], 1.056555029, 11.00333771, [7/9, 2/9, 1/3]] [`2F1(1,16/9;10/3;-1/9)`, .9446549057807403927320643814063932672259104891099062\ 530975283885136842679056232706333340058168494433, 3*(3*n+8)*(9*n+14)*(9*n+16)/( n+2)/(3*n+5)/(10+3*n)-(6*n+13)*(513*n^2+2223*n+2282)/(n+2)/(3*n+5)/(10+3*n)*N+N ^2, [[0, 270], [1, 286]], 1.148873176, 25.01345168, [7/9, 5/9, 1/9]] [`2F1(1,16/9;32/9;-1/9)`, .9479452159876181648421839230971397668878650482214696\ 021225275903195907103104073052492639236364290846, 9*(9*n+25)*(9*n+16)/(32+9*n)/ (n+2)-19*(9*n+25)*(18*n+41)/(32+9*n)/(n+2)*N+N^2, [[0, 288], [1, 304]], .843023\ 1414, 12.19114502, [7/9, 7/9, 1/9]] [`2F1(1,16/9;11/3;-1/9)`, .9494463859196704855427382313667794383514446351694483\ 222913584730151989939453861513064616542569787942, 3*(9*n+16)*(9*n+17)*(6*n+17)/ (n+2)/(6*n+11)/(11+3*n)-2*(3*n+7)*(1026*n^2+4788*n+5327)/(n+2)/(6*n+11)/(11+3*n )*N+N^2, [[0, 297], [1, 313]], 1.157707907, 25.79720448, [7/9, 8/9, 1/9]] [`2F1(1,17/9;3;-1/9)`, .9352101510729855067512044976486162385484582592592703744\ 147724529458792982292361570824456366998375475, (9*n+17)*(5+2*n)*(9*n+10)/(n+2)/ (2*n+3)/(3+n)-4*(171*n^2+684*n+643)/(3+n)/(2*n+3)*N+N^2, [[0, 243], [1, 260]], -.4111355898e-1, 4.912125579, [8/9, 1/9, 1/9]] [`2F1(1,17/9;3;-2/9)`, .8792494078156863906116058519557237740011313503474900400\ 490850877806251031448281166136655893601694396, 1/4*(9*n+17)*(5+2*n)*(9*n+10)/(n +2)/(2*n+3)/(3+n)-1/2*(360*n^2+1440*n+1357)/(3+n)/(2*n+3)*N+N^2, [[0, 243/4], [ 1, 277/4]], 1.346229019, 41.88774831, [8/9, 1/9, 2/9]] [`2F1(1,17/9;3;-1/3)`, .8303319223520689717453129011689070910168384441778161974\ 307505196228745801350273263063731650449697629, 1/9*(9*n+17)*(5+2*n)*(9*n+10)/(n +2)/(2*n+3)/(3+n)-28/3*(9*n^2+36*n+34)/(3+n)/(2*n+3)*N+N^2, [[0, 27], [1, 98/3] ], 1.058688574, 11.05259364, [8/9, 1/9, 1/3]] [`2F1(1,17/9;10/3;-1/9)`, .9413528565664306177854788794400723745144313575711293\ 828221694674987439169167626289002382556915605819, 3*(9*n+17)*(9*n+13)*(3*n+8)/( n+2)/(3*n+5)/(10+3*n)-(6*n+13)*(513*n^2+2223*n+2284)/(n+2)/(3*n+5)/(10+3*n)*N+N ^2, [[0, 270], [1, 287]], 1.142256776, 24.45698671, [8/9, 4/9, 1/9]] [`2F1(1,17/9;11/3;-1/9)`, .9464203510049951951261713666175762852618772036994879\ 675414324566520755255342745179843319018539243604, 3*(9*n+16)*(9*n+17)*(6*n+17)/ (n+2)/(6*n+11)/(11+3*n)-4*(3*n+7)*(513*n^2+2394*n+2666)/(n+2)/(6*n+11)/(11+3*n) *N+N^2, [[0, 297], [1, 314]], 1.155944850, 25.63690060, [8/9, 7/9, 1/9]] [`2F1(1,17/9;34/9;-1/9)`, .9479183654223170276950886677316905089597685075559468\ 174519346122500100304895752611080345947114620623, 9*(9*n+26)*(9*n+17)/(34+9*n)/ (n+2)-19*(9*n+26)*(18*n+43)/(34+9*n)/(n+2)*N+N^2, [[0, 306], [1, 323]], .818113\ 1708, 11.70256222, [8/9, 8/9, 1/9]] ----------------------- --------------------------------- This took, 1407.646, seconds .