Diophantine Approximations to the Constant, 8 - 8 Catalan, in terms of ratios of sequences satisfying the same recurrence By Shalosh B. Ekhad Let a(n), b(n) be the sequences satisfying the linear recurrence 12 -4 (4 n + 7) (2 n + 5) (2 n + 3) (4 n + 5) (n + 2) (52264960 n 11 10 9 8 + 2030977024 n + 36030949376 n + 385846651904 n + 2777570971264 n 7 6 5 + 14158394250432 n + 52396463329552 n + 141826798615504 n 4 3 2 + 278645102200192 n + 387470620438796 n + 361936752097629 n 3 + 203887749751029 n + 52375197579606) (n + 1) x(n) - (2 n + 5) (n + 2) ( 18 17 16 93658808320 n + 4442300612608 n + 98609975787520 n 15 14 13 + 1360831749554176 n + 13079532286623744 n + 92951541658091520 n 12 11 + 506035285156910080 n + 2157227191810272256 n 10 9 + 7297982136176020096 n + 19733691684321877312 n 8 7 + 42737943644526475488 n + 73936531239268222224 n 6 5 + 101386504389862999144 n + 108672483675323635180 n 4 3 + 89020897591196145550 n + 53760334869617318227 n 2 + 22532396006906359359 n + 5847183480905485308 n + 706831279248516660) 20 19 18 x(n + 1) + (-374635233280 n - 19736037425152 n - 489994598219776 n 17 16 15 - 7620735203016704 n - 83239973104549888 n - 678496390560268288 n 14 13 - 4280355503556986880 n - 21390040017391047680 n 12 11 - 85946830403016530688 n - 280230257724154001664 n 10 9 - 744921855855156628416 n - 1615841166585277569920 n 8 7 - 2852139023458678346192 n - 4069319600262292293120 n 6 5 - 4640186331720693371516 n - 4155868720005903566500 n 4 3 - 2848100554339460545316 n - 1434835030172870395815 n 2 - 497697248310778866051 n - 105304030601373610110 n 19 - 10118957751250084800) x(n + 2) + (2 n + 7) (314425999360 n 18 17 16 + 17018371702784 n + 433467110981632 n + 6904714503913472 n 15 14 13 + 77098494223368192 n + 641050936773640192 n + 4115108417504090112 n 12 11 + 20865093236832951296 n + 84776511535262013440 n 10 9 + 278381831158711441408 n + 741617762990343018048 n 8 7 + 1602349368669267321728 n + 2795346107523440017616 n 6 5 + 3901631834529069184640 n + 4291854268476187629708 n 4 3 + 3634464037444094034384 n + 2283750156853067922477 n 2 + 1001705654861635954131 n + 273437324359499819610 n 12 + 34929204730102009800) x(n + 3) - (2 n + 9) (2 n + 7) (52264960 n 11 10 9 8 + 1403797504 n + 17139689472 n + 125742603264 n + 617103772288 n 7 6 5 + 2133420593344 n + 5325052818896 n + 9664119940400 n 4 3 2 + 12649698255648 n + 11639773328940 n + 7142834711369 n 2 2 2 + 2623127095631 n + 435728707890) (4 n + 15) (n + 4) (4 n + 17) x(n + 4) = 0 and in Maple format -4*(4*n+7)*(2*n+5)*(2*n+3)*(4*n+5)*(n+2)*(52264960*n^12+2030977024*n^11+ 36030949376*n^10+385846651904*n^9+2777570971264*n^8+14158394250432*n^7+ 52396463329552*n^6+141826798615504*n^5+278645102200192*n^4+387470620438796*n^3+ 361936752097629*n^2+203887749751029*n+52375197579606)*(n+1)^3*x(n)-(2*n+5)*(n+2 )*(93658808320*n^18+4442300612608*n^17+98609975787520*n^16+1360831749554176*n^ 15+13079532286623744*n^14+92951541658091520*n^13+506035285156910080*n^12+ 2157227191810272256*n^11+7297982136176020096*n^10+19733691684321877312*n^9+ 42737943644526475488*n^8+73936531239268222224*n^7+101386504389862999144*n^6+ 108672483675323635180*n^5+89020897591196145550*n^4+53760334869617318227*n^3+ 22532396006906359359*n^2+5847183480905485308*n+706831279248516660)*x(n+1)+(-\ 374635233280*n^20-19736037425152*n^19-489994598219776*n^18-7620735203016704*n^ 17-83239973104549888*n^16-678496390560268288*n^15-4280355503556986880*n^14-\ 21390040017391047680*n^13-85946830403016530688*n^12-280230257724154001664*n^11-\ 744921855855156628416*n^10-1615841166585277569920*n^9-2852139023458678346192*n^ 8-4069319600262292293120*n^7-4640186331720693371516*n^6-4155868720005903566500* n^5-2848100554339460545316*n^4-1434835030172870395815*n^3-497697248310778866051 *n^2-105304030601373610110*n-10118957751250084800)*x(n+2)+(2*n+7)*(314425999360 *n^19+17018371702784*n^18+433467110981632*n^17+6904714503913472*n^16+ 77098494223368192*n^15+641050936773640192*n^14+4115108417504090112*n^13+ 20865093236832951296*n^12+84776511535262013440*n^11+278381831158711441408*n^10+ 741617762990343018048*n^9+1602349368669267321728*n^8+2795346107523440017616*n^7 +3901631834529069184640*n^6+4291854268476187629708*n^5+3634464037444094034384*n ^4+2283750156853067922477*n^3+1001705654861635954131*n^2+273437324359499819610* n+34929204730102009800)*x(n+3)-(2*n+9)*(2*n+7)*(52264960*n^12+1403797504*n^11+ 17139689472*n^10+125742603264*n^9+617103772288*n^8+2133420593344*n^7+ 5325052818896*n^6+9664119940400*n^5+12649698255648*n^4+11639773328940*n^3+ 7142834711369*n^2+2623127095631*n+435728707890)*(4*n+15)^2*(n+4)^2*(4*n+17)^2*x (n+4) = 0 subject to the initial conditions 32 2872 9539264 2049157592 a(1) = --, a(2) = ----, a(3) = -------, a(4) = ---------- 15 225 96525 2436525 b(1) = 3, b(2) = 19, b(3) = 147, b(4) = 1251 Then the limit a(n)/b(n), as n goes to infinity is indeed the constant, 8 - 8 Catalan The delta such that the error is less than the reciporocal of the denominato\ r to the power delta+1 is approximately, -0.7024891810 This is very disappointing ---------------------------------------- 1/2 1/2 Diophantine Approximations to the Constant, 3 I dilog(1/2 - 1/2 I 3 ) 3 2 1/2 1/2 1/2 Pi - 3 I dilog(1/2 + 1/2 I 3 ) 3 + 18 - 3 dilog(1/2 - 1/2 I 3 ) - --- 2 1/2 - 3 dilog(1/2 + 1/2 I 3 ), in terms of ratios of sequences satisfying the same recurrence By Shalosh B. Ekhad Let a(n), b(n) be the sequences satisfying the linear recurrence 12 -9 (3 n + 5) (3 n + 7) (3 n + 4) (6 n + 7) (n + 2) (376732620 n 11 10 9 + 14353867116 n + 249662957697 n + 2621088717975 n 8 7 6 + 18496760057964 n + 92423781148284 n + 335264333081391 n 5 4 3 + 889480284352425 n + 1712771814114945 n + 2334181262572317 n 2 3 + 2136750360031426 n + 1179545645197472 n + 296912338379840) (n + 1) 18 17 x(n) - (3 n + 7) (n + 2) (2563288746480 n + 118963420726824 n 16 15 14 + 2583702553831380 n + 34882428671667774 n + 327978643459471524 n 13 12 + 2280020532178095540 n + 12141666702183849039 n 11 10 + 50629751422155688923 n + 167546221523260154775 n 9 8 + 443186261311617289674 n + 939028725126128834892 n 7 6 + 1589543449326384417045 n + 2133165685627876900095 n 5 4 + 2238222216243389377716 n + 1795356349539865201443 n 3 2 + 1062092559730396853152 n + 436264418096170761460 n + 111011624088674755800 n + 13167254285785138000) x(n + 1) + ( 20 19 18 -15379732478880 n - 797270500674864 n - 19490252808945672 n 17 16 - 298692567369502272 n - 3217629159487349316 n 15 14 - 25892114911359659307 n - 161447687491608558747 n 13 12 - 798553086240906480312 n - 3181134703719963277341 n 11 10 - 10303466494493419092423 n - 27272163085124033473551 n 9 8 - 59072091291019024790787 n - 104478896235283561479933 n 7 6 - 150000605485048428694614 n - 173022157363350075444702 n 5 4 - 157795001963245739139669 n - 111051016429719965115090 n 3 2 - 58089516580802781278312 n - 21233134561080204415168 n - 4832374540452125228240 n - 514613488835214501600) x(n + 2) + (3 n + 10) 19 18 17 (8605326506040 n + 455211555747732 n + 11329680346533990 n 16 15 + 176314511183615955 n + 1922995016456165091 n 14 13 + 15614141946219983970 n + 97858288708817238558 n 12 11 + 484304620635661879953 n + 1920173285526106321233 n 10 9 + 6151061988356330314656 n + 15981003791320868305281 n 8 7 + 33663428387340230717631 n + 57235811568468098691366 n 6 5 + 77831289923901127944462 n + 83380170567366422082267 n 4 3 + 68737010886651687285233 n + 42028200888061937414334 n 2 + 17929529048722418251368 n + 4757740301861954883280 n 12 + 590478574893149023200) x(n + 3) - 2 (3 n + 13) (3 n + 10) (376732620 n 11 10 9 8 + 9833075676 n + 116634772341 n + 831040655985 n + 3959889265314 n 7 6 5 + 13287743521272 n + 32181529806153 n + 56650354229817 n 4 3 2 + 71898001591365 n + 64121369236923 n + 38120528120593 n 2 2 2 + 13555807871487 n + 2179229500294) (6 n + 25) (n + 4) (3 n + 11) x(n + 4) = 0 and in Maple format -9*(3*n+5)*(3*n+7)*(3*n+4)*(6*n+7)*(n+2)*(376732620*n^12+14353867116*n^11+ 249662957697*n^10+2621088717975*n^9+18496760057964*n^8+92423781148284*n^7+ 335264333081391*n^6+889480284352425*n^5+1712771814114945*n^4+2334181262572317*n ^3+2136750360031426*n^2+1179545645197472*n+296912338379840)*(n+1)^3*x(n)-(3*n+7 )*(n+2)*(2563288746480*n^18+118963420726824*n^17+2583702553831380*n^16+ 34882428671667774*n^15+327978643459471524*n^14+2280020532178095540*n^13+ 12141666702183849039*n^12+50629751422155688923*n^11+167546221523260154775*n^10+ 443186261311617289674*n^9+939028725126128834892*n^8+1589543449326384417045*n^7+ 2133165685627876900095*n^6+2238222216243389377716*n^5+1795356349539865201443*n^ 4+1062092559730396853152*n^3+436264418096170761460*n^2+111011624088674755800*n+ 13167254285785138000)*x(n+1)+(-15379732478880*n^20-797270500674864*n^19-\ 19490252808945672*n^18-298692567369502272*n^17-3217629159487349316*n^16-\ 25892114911359659307*n^15-161447687491608558747*n^14-798553086240906480312*n^13 -3181134703719963277341*n^12-10303466494493419092423*n^11-\ 27272163085124033473551*n^10-59072091291019024790787*n^9-\ 104478896235283561479933*n^8-150000605485048428694614*n^7-\ 173022157363350075444702*n^6-157795001963245739139669*n^5-\ 111051016429719965115090*n^4-58089516580802781278312*n^3-\ 21233134561080204415168*n^2-4832374540452125228240*n-514613488835214501600)*x(n +2)+(3*n+10)*(8605326506040*n^19+455211555747732*n^18+11329680346533990*n^17+ 176314511183615955*n^16+1922995016456165091*n^15+15614141946219983970*n^14+ 97858288708817238558*n^13+484304620635661879953*n^12+1920173285526106321233*n^ 11+6151061988356330314656*n^10+15981003791320868305281*n^9+ 33663428387340230717631*n^8+57235811568468098691366*n^7+77831289923901127944462 *n^6+83380170567366422082267*n^5+68737010886651687285233*n^4+ 42028200888061937414334*n^3+17929529048722418251368*n^2+4757740301861954883280* n+590478574893149023200)*x(n+3)-2*(3*n+13)*(3*n+10)*(376732620*n^12+9833075676* n^11+116634772341*n^10+831040655985*n^9+3959889265314*n^8+13287743521272*n^7+ 32181529806153*n^6+56650354229817*n^5+71898001591365*n^4+64121369236923*n^3+ 38120528120593*n^2+13555807871487*n+2179229500294)*(6*n+25)^2*(n+4)^2*(3*n+11)^ 2*x(n+4) = 0 subject to the initial conditions 153 422361 88722981 1511583500313 a(1) = ---, a(2) = ------, a(3) = --------, a(4) = ------------- 56 25480 691600 1384583200 b(1) = 3, b(2) = 19, b(3) = 147, b(4) = 1251 Then the limit a(n)/b(n), as n goes to infinity is indeed the constant, 1/2 1/2 1/2 1/2 3 I dilog(1/2 - 1/2 I 3 ) 3 - 3 I dilog(1/2 + 1/2 I 3 ) 3 + 18 2 1/2 Pi 1/2 - 3 dilog(1/2 - 1/2 I 3 ) - --- - 3 dilog(1/2 + 1/2 I 3 ) 2 The delta such that the error is less than the reciporocal of the denominato\ r to the power delta+1 is approximately, -0.7309904133 This is very disappointing ---------------------------------------- 1/2 1/2 Diophantine Approximations to the Constant, 3 I dilog(1/2 - 1/2 I 3 ) 3 2 1/2 1/2 1/2 Pi - 3 I dilog(1/2 + 1/2 I 3 ) 3 + 9/2 + 3 dilog(1/2 - 1/2 I 3 ) + --- 2 1/2 + 3 dilog(1/2 + 1/2 I 3 ), in terms of ratios of sequences satisfying the same recurrence By Shalosh B. Ekhad Let a(n), b(n) be the sequences satisfying the linear recurrence 12 -9 (6 n + 11) (3 n + 8) (3 n + 5) (3 n + 4) (n + 2) (376732620 n 11 10 9 + 14925225240 n + 269956484991 n + 2947431213351 n 8 7 6 + 21632851238553 n + 112432371142158 n + 424246098137607 n 5 4 3 + 1170909686250981 n + 2345728518796293 n + 3326136715078566 n 2 3 + 3168290075385040 n + 1820090648257112 n + 476825285163632) (n + 1) 18 17 x(n) - (3 n + 8) (n + 2) (2563288746480 n + 124193616460200 n 16 15 14 + 2816240921541504 n + 39702686293205694 n + 389830070506432257 n 13 12 + 2830082606020477332 n + 15738487165882642041 n 11 10 + 68530542165853377111 n + 236781057487127370561 n 9 8 + 653789958268492544535 n + 1445557803520480249725 n 7 6 + 2552413292161672527135 n + 3570978768997991234994 n 5 4 + 3903445927819550680209 n + 3259137251326432641294 n 3 2 + 2004757015786697252312 n + 855144996316607673560 n + 225620909020745183376 n + 27696446235536775488) x(n + 1) + ( 20 19 18 -15379732478880 n - 823158913475520 n - 20746842211299204 n 17 16 - 327251053036121832 n - 3621125943623266605 n 15 14 - 29859884852507738964 n - 190248470993553387021 n 13 12 - 958208139266974393275 n - 3870578239024201380729 n 11 10 - 12646326536426210553099 n - 33549299578590484598433 n 9 8 - 72240367202139070924605 n - 125684377674925068697878 n 7 6 - 175044790981586858014629 n - 192179490937567270886034 n 5 4 - 162368747160959223671436 n - 101623291588472594029536 n 3 2 - 44185351933678686106448 n - 11780705147513289923968 n - 1371465919588951680512 n + 36209649591148627968) x(n + 2) + (3 n + 11) ( 19 18 17 8605326506040 n + 476318568362220 n + 12408781039410402 n 16 15 + 202199221365737625 n + 2310002582168485476 n 14 13 + 19654814640712832757 n + 129136758190026455223 n 12 11 + 670299875083659854529 n + 2788680201138628922043 n 10 9 + 9378626257050103196001 n + 25595389344130280875791 n 8 7 + 56668051976969481524946 n + 101330762504923607403069 n 6 5 + 145014702234065468767746 n + 163612956298158418101108 n 4 3 + 142160945428652207876752 n + 91691964145048265658192 n 2 + 41300843299328889014016 n + 11583152090474872971776 n + 1521058761335006742528) x(n + 3) - 2 (3 n + 14) (3 n + 11) ( 12 11 10 9 376732620 n + 10404433800 n + 130643360271 n + 985872575241 n 8 7 6 + 4977832625289 n + 17709258809610 n + 45498621628611 n 5 4 3 + 85017602542023 n + 114613843612257 n + 108660084978390 n 2 2 + 68730188284240 n + 26029170809952 n + 4461384771328) (n + 4) 2 2 (6 n + 23) (3 n + 13) x(n + 4) = 0 and in Maple format -9*(6*n+11)*(3*n+8)*(3*n+5)*(3*n+4)*(n+2)*(376732620*n^12+14925225240*n^11+ 269956484991*n^10+2947431213351*n^9+21632851238553*n^8+112432371142158*n^7+ 424246098137607*n^6+1170909686250981*n^5+2345728518796293*n^4+3326136715078566* n^3+3168290075385040*n^2+1820090648257112*n+476825285163632)*(n+1)^3*x(n)-(3*n+ 8)*(n+2)*(2563288746480*n^18+124193616460200*n^17+2816240921541504*n^16+ 39702686293205694*n^15+389830070506432257*n^14+2830082606020477332*n^13+ 15738487165882642041*n^12+68530542165853377111*n^11+236781057487127370561*n^10+ 653789958268492544535*n^9+1445557803520480249725*n^8+2552413292161672527135*n^7 +3570978768997991234994*n^6+3903445927819550680209*n^5+3259137251326432641294*n ^4+2004757015786697252312*n^3+855144996316607673560*n^2+225620909020745183376*n +27696446235536775488)*x(n+1)+(-15379732478880*n^20-823158913475520*n^19-\ 20746842211299204*n^18-327251053036121832*n^17-3621125943623266605*n^16-\ 29859884852507738964*n^15-190248470993553387021*n^14-958208139266974393275*n^13 -3870578239024201380729*n^12-12646326536426210553099*n^11-\ 33549299578590484598433*n^10-72240367202139070924605*n^9-\ 125684377674925068697878*n^8-175044790981586858014629*n^7-\ 192179490937567270886034*n^6-162368747160959223671436*n^5-\ 101623291588472594029536*n^4-44185351933678686106448*n^3-\ 11780705147513289923968*n^2-1371465919588951680512*n+36209649591148627968)*x(n+ 2)+(3*n+11)*(8605326506040*n^19+476318568362220*n^18+12408781039410402*n^17+ 202199221365737625*n^16+2310002582168485476*n^15+19654814640712832757*n^14+ 129136758190026455223*n^13+670299875083659854529*n^12+2788680201138628922043*n^ 11+9378626257050103196001*n^10+25595389344130280875791*n^9+ 56668051976969481524946*n^8+101330762504923607403069*n^7+ 145014702234065468767746*n^6+163612956298158418101108*n^5+ 142160945428652207876752*n^4+91691964145048265658192*n^3+ 41300843299328889014016*n^2+11583152090474872971776*n+1521058761335006742528)*x (n+3)-2*(3*n+14)*(3*n+11)*(376732620*n^12+10404433800*n^11+130643360271*n^10+ 985872575241*n^9+4977832625289*n^8+17709258809610*n^7+45498621628611*n^6+ 85017602542023*n^5+114613843612257*n^4+108660084978390*n^3+68730188284240*n^2+ 26029170809952*n+4461384771328)*(n+4)^2*(6*n+23)^2*(3*n+13)^2*x(n+4) = 0 subject to the initial conditions 171 622053 128733849 16050398455197 a(1) = ---, a(2) = ------, a(3) = ---------, a(4) = -------------- 100 61600 1645600 24109685600 b(1) = 3, b(2) = 19, b(3) = 147, b(4) = 1251 Then the limit a(n)/b(n), as n goes to infinity is indeed the constant, 1/2 1/2 1/2 1/2 3 I dilog(1/2 - 1/2 I 3 ) 3 - 3 I dilog(1/2 + 1/2 I 3 ) 3 + 9/2 2 1/2 Pi 1/2 + 3 dilog(1/2 - 1/2 I 3 ) + --- + 3 dilog(1/2 + 1/2 I 3 ) 2 The delta such that the error is less than the reciporocal of the denominato\ r to the power delta+1 is approximately, -0.7287758729 This is very disappointing ---------------------------------------- Diophantine Approximations to the Constant, 1/2 1/2 2 1/2 4 I 2 dilog(1 - ---- - 1/2 I 2 ) 2 1/2 1/2 2 1/2 + 4 I 2 dilog(1 + ---- - 1/2 I 2 ) 2 1/2 1/2 2 1/2 - 4 I 2 dilog(1 + ---- + 1/2 I 2 ) 2 1/2 1/2 2 1/2 - 4 I 2 dilog(1 - ---- + 1/2 I 2 ) 2 1/2 1/2 2 1/2 - 4 2 dilog(1 - ---- - 1/2 I 2 ) 2 1/2 1/2 2 1/2 + 4 2 dilog(1 + ---- - 1/2 I 2 ) 2 1/2 1/2 2 1/2 + 4 2 dilog(1 + ---- + 1/2 I 2 ) 2 1/2 1/2 2 1/2 - 4 2 dilog(1 - ---- + 1/2 I 2 ) + 32, 2 in terms of ratios of sequences satisfying the same recurrence By Shalosh B. Ekhad Let a(n), b(n) be the sequences satisfying the linear recurrence 12 -16 (8 n + 13) (4 n + 9) (4 n + 5) (8 n + 9) (n + 2) (53519319040 n 11 10 9 + 2018844344320 n + 34763523489792 n + 361299236093952 n 8 7 6 + 2523929716596736 n + 12483691433670656 n + 44823588073512704 n 5 4 3 + 117706028344741632 n + 224330861906074496 n + 302577142294610472 n 2 + 274127774187407055 n + 149760797798033739 n + 37306354425093054) 3 18 (n + 1) x(n) - (4 n + 9) (n + 2) (3069011831029760 n 17 16 + 140868742555893760 n + 3025611381924691968 n 15 14 + 40394092608065896448 n + 375552384795624341504 n 13 12 + 2581389639256652120064 n + 13591339669262024835072 n 11 10 + 56032773331673651085312 n + 183320533655212808159232 n 9 8 + 479397763933873450536960 n + 1004201740343117695360000 n 7 6 + 1680555247427441323872000 n + 2229752561373263241300928 n 5 4 + 2313174655106905143580208 n + 1834674205216140243219468 n 3 2 + 1073278326616631649883609 n + 436002328809288460459995 n + 109736804756042627933052 n + 12876207480328528897236) x(n + 1) + ( 20 19 18 -24552094648238080 n - 1262424891274035200 n - 30618620994743435264 n 17 16 - 465677212105658335232 n - 4980020243655158661120 n 15 14 - 39797866002268746153984 n - 246551186169558483337216 n 13 12 - 1212200513376350232903680 n - 4802759256527999072927744 n 11 10 - 15481303560611137354072064 n - 40810896184500760452599808 n 9 8 - 88111728667488105489500160 n - 155485322846554590543040768 n 7 6 - 222965571177540998570345984 n - 257201687297191076848200464 n 5 4 - 234917283915611716758250736 n - 165846754914033515010427956 n 3 2 - 87189090667062591491222643 n - 32099277601431830700168723 n - 7376201366106371137124742 n - 795384641531482242376032) x(n + 2) + 19 18 (4 n + 13) (10303111147028480 n + 538704519438008320 n 17 16 + 13250859715384049664 n + 203776103766577643520 n 15 14 + 2195982027697676091392 n + 17615606200329215934464 n 13 12 + 109055117975927789715456 n + 533055869432087375249408 n 11 10 + 2087052781800551101956096 n + 6601028246329968958881792 n 9 8 + 16930112521937549939464192 n + 35199000682156197843261952 n 7 6 + 59057328959483584613671168 n + 79233260256238782262050432 n 5 4 + 83728263755999465300828136 n + 68070588542572804587226314 n 3 2 + 41036201304989750933315049 n + 17256171032079232740085737 n + 4512428064159114987659502 n + 551727801627109016874696) x(n + 3) - 12 11 (4 n + 17) (4 n + 13) (53519319040 n + 1376612515840 n 10 9 8 + 16088510758912 n + 112926189944832 n + 529977894903808 n 7 6 5 + 1751234714449920 n + 4175729967208192 n + 7235570496138496 n 4 3 2 + 9037330882673024 n + 7930200776271400 n + 4637650246699095 n 2 2 2 + 1621867214212389 n + 256347399998106) (n + 4) (8 n + 29) (8 n + 33) x(n + 4) = 0 and in Maple format -16*(8*n+13)*(4*n+9)*(4*n+5)*(8*n+9)*(n+2)*(53519319040*n^12+2018844344320*n^11 +34763523489792*n^10+361299236093952*n^9+2523929716596736*n^8+12483691433670656 *n^7+44823588073512704*n^6+117706028344741632*n^5+224330861906074496*n^4+ 302577142294610472*n^3+274127774187407055*n^2+149760797798033739*n+ 37306354425093054)*(n+1)^3*x(n)-(4*n+9)*(n+2)*(3069011831029760*n^18+ 140868742555893760*n^17+3025611381924691968*n^16+40394092608065896448*n^15+ 375552384795624341504*n^14+2581389639256652120064*n^13+13591339669262024835072* n^12+56032773331673651085312*n^11+183320533655212808159232*n^10+ 479397763933873450536960*n^9+1004201740343117695360000*n^8+ 1680555247427441323872000*n^7+2229752561373263241300928*n^6+ 2313174655106905143580208*n^5+1834674205216140243219468*n^4+ 1073278326616631649883609*n^3+436002328809288460459995*n^2+ 109736804756042627933052*n+12876207480328528897236)*x(n+1)+(-24552094648238080* n^20-1262424891274035200*n^19-30618620994743435264*n^18-465677212105658335232*n ^17-4980020243655158661120*n^16-39797866002268746153984*n^15-\ 246551186169558483337216*n^14-1212200513376350232903680*n^13-\ 4802759256527999072927744*n^12-15481303560611137354072064*n^11-\ 40810896184500760452599808*n^10-88111728667488105489500160*n^9-\ 155485322846554590543040768*n^8-222965571177540998570345984*n^7-\ 257201687297191076848200464*n^6-234917283915611716758250736*n^5-\ 165846754914033515010427956*n^4-87189090667062591491222643*n^3-\ 32099277601431830700168723*n^2-7376201366106371137124742*n-\ 795384641531482242376032)*x(n+2)+(4*n+13)*(10303111147028480*n^19+ 538704519438008320*n^18+13250859715384049664*n^17+203776103766577643520*n^16+ 2195982027697676091392*n^15+17615606200329215934464*n^14+ 109055117975927789715456*n^13+533055869432087375249408*n^12+ 2087052781800551101956096*n^11+6601028246329968958881792*n^10+ 16930112521937549939464192*n^9+35199000682156197843261952*n^8+ 59057328959483584613671168*n^7+79233260256238782262050432*n^6+ 83728263755999465300828136*n^5+68070588542572804587226314*n^4+ 41036201304989750933315049*n^3+17256171032079232740085737*n^2+ 4512428064159114987659502*n+551727801627109016874696)*x(n+3)-(4*n+17)*(4*n+13)* (53519319040*n^12+1376612515840*n^11+16088510758912*n^10+112926189944832*n^9+ 529977894903808*n^8+1751234714449920*n^7+4175729967208192*n^6+7235570496138496* n^5+9037330882673024*n^4+7930200776271400*n^3+4637650246699095*n^2+ 1621867214212389*n+256347399998106)*(n+4)^2*(8*n+29)^2*(8*n+33)^2*x(n+4) = 0 subject to the initial conditions 704 8557088 14878272 3433169900192 a(1) = ---, a(2) = -------, a(3) = --------, a(4) = ------------- 225 447525 100555 2726548825 b(1) = 3, b(2) = 19, b(3) = 147, b(4) = 1251 Then the limit a(n)/b(n), as n goes to infinity is indeed the constant, 1/2 1/2 2 1/2 4 I 2 dilog(1 - ---- - 1/2 I 2 ) 2 1/2 1/2 2 1/2 + 4 I 2 dilog(1 + ---- - 1/2 I 2 ) 2 1/2 1/2 2 1/2 - 4 I 2 dilog(1 + ---- + 1/2 I 2 ) 2 1/2 1/2 2 1/2 - 4 I 2 dilog(1 - ---- + 1/2 I 2 ) 2 1/2 1/2 2 1/2 - 4 2 dilog(1 - ---- - 1/2 I 2 ) 2 1/2 1/2 2 1/2 + 4 2 dilog(1 + ---- - 1/2 I 2 ) 2 1/2 1/2 2 1/2 + 4 2 dilog(1 + ---- + 1/2 I 2 ) 2 1/2 1/2 2 1/2 - 4 2 dilog(1 - ---- + 1/2 I 2 ) + 32 2 The delta such that the error is less than the reciporocal of the denominato\ r to the power delta+1 is approximately, -0.7626238194 This is very disappointing ---------------------------------------- Diophantine Approximations to the Constant, 1/2 1/2 2 1/2 4 I 2 dilog(1 - ---- - 1/2 I 2 ) 2 1/2 1/2 2 1/2 + 4 I 2 dilog(1 + ---- - 1/2 I 2 ) 2 1/2 1/2 2 1/2 - 4 I 2 dilog(1 + ---- + 1/2 I 2 ) 2 1/2 1/2 2 1/2 - 4 I 2 dilog(1 - ---- + 1/2 I 2 ) + 32/9 2 1/2 1/2 2 1/2 + 4 2 dilog(1 - ---- - 1/2 I 2 ) 2 1/2 1/2 2 1/2 - 4 2 dilog(1 + ---- - 1/2 I 2 ) 2 1/2 1/2 2 1/2 - 4 2 dilog(1 + ---- + 1/2 I 2 ) 2 1/2 1/2 2 1/2 + 4 2 dilog(1 - ---- + 1/2 I 2 ), 2 in terms of ratios of sequences satisfying the same recurrence By Shalosh B. Ekhad Let a(n), b(n) be the sequences satisfying the linear recurrence 12 -16 (8 n + 15) (4 n + 11) (4 n + 7) (8 n + 11) (n + 2) (53519319040 n 11 10 9 + 2140596600832 n + 39087926411264 n + 430849919549440 n 8 7 6 + 3192482338971648 n + 16750895801346048 n + 63811181749382912 n 5 4 3 + 177801724601173760 n + 359605747225928320 n + 514789298293895864 n 2 + 495063600588959247 n + 287133627748496859 n + 75947986423062990) 3 18 (n + 1) x(n) - (4 n + 11) (n + 2) (3069011831029760 n 17 16 + 150261870391984128 n + 3443236787558809600 n 15 14 + 49052373798478675968 n + 486686825575966834688 n 13 12 + 3570204796102031441920 n + 20061147955400178925568 n 11 10 + 88256494460975110946816 n + 308063513762197090549760 n 9 8 + 859234144068608322367488 n + 1918772373848554879089664 n 7 6 + 3421141350890451132181760 n + 4832096197426488015911360 n 5 4 + 5330838552655358185739152 n + 4490439885228936246194796 n 3 2 + 2785407832394563271560779 n + 1197464770331128451769705 n + 318195974352120930267300 n + 39305481509098754581500) x(n + 1) + ( 20 19 18 -24552094648238080 n - 1324417006115487744 n - 33627636461975109632 n 17 16 - 534058323565663485952 n - 5945950322387998212096 n 15 14 - 49291694612341852209152 n - 315400601988183361585152 n 13 12 - 1593250087622050981609472 n - 6443836310474030035173376 n 11 10 - 21033288218582573103710208 n - 55576747262252046136119296 n 9 8 - 118697529465947346922289152 n - 203600243226843972000366848 n 7 6 - 277024665807852751135742464 n - 292777149238353216410433552 n 5 4 - 231953324821757706210900464 n - 128987377247403125517753220 n 3 2 - 43137973095293363182106547 n - 3864366879715031206661523 n + 2711175096050231711437530 n + 792490345419546744141600) x(n + 2) + 19 18 (4 n + 15) (10303111147028480 n + 576611488475643904 n 17 16 + 15188861899799265280 n + 250272710903883366400 n 15 14 + 2891439393774535442432 n + 24881134691737004933120 n 13 12 + 165342758808532645052416 n + 868111830671527526793216 n 11 10 + 3653563119674999706157056 n + 12431167113957899989696512 n 9 8 + 34327032710530040730867712 n + 76906998164751357268936192 n 7 6 + 139180322823943605263223552 n + 201613120410950857929225344 n 5 4 + 230283144961655607745023464 n + 202598927014753259443433366 n 3 2 + 132337694461799072866880883 n + 60380898591740320609115931 n + 17157746740315033962082890 n + 2283444201992095232818200) x(n + 3) - 12 11 (4 n + 19) (4 n + 15) (53519319040 n + 1498364772352 n 10 9 8 + 19073638858752 n + 145929218293760 n + 747083375321088 n 7 6 5 + 2695092601595904 n + 7021984156587776 n + 13307935687226624 n 4 3 2 + 18198731009493120 n + 17504528740711096 n + 11235522001524711 n 2 2 2 + 4318951298386149 n + 751602810972618) (8 n + 35) (n + 4) (8 n + 31) x(n + 4) = 0 and in Maple format -16*(8*n+15)*(4*n+11)*(4*n+7)*(8*n+11)*(n+2)*(53519319040*n^12+2140596600832*n^ 11+39087926411264*n^10+430849919549440*n^9+3192482338971648*n^8+ 16750895801346048*n^7+63811181749382912*n^6+177801724601173760*n^5+ 359605747225928320*n^4+514789298293895864*n^3+495063600588959247*n^2+ 287133627748496859*n+75947986423062990)*(n+1)^3*x(n)-(4*n+11)*(n+2)*( 3069011831029760*n^18+150261870391984128*n^17+3443236787558809600*n^16+ 49052373798478675968*n^15+486686825575966834688*n^14+3570204796102031441920*n^ 13+20061147955400178925568*n^12+88256494460975110946816*n^11+ 308063513762197090549760*n^10+859234144068608322367488*n^9+ 1918772373848554879089664*n^8+3421141350890451132181760*n^7+ 4832096197426488015911360*n^6+5330838552655358185739152*n^5+ 4490439885228936246194796*n^4+2785407832394563271560779*n^3+ 1197464770331128451769705*n^2+318195974352120930267300*n+ 39305481509098754581500)*x(n+1)+(-24552094648238080*n^20-1324417006115487744*n^ 19-33627636461975109632*n^18-534058323565663485952*n^17-5945950322387998212096* n^16-49291694612341852209152*n^15-315400601988183361585152*n^14-\ 1593250087622050981609472*n^13-6443836310474030035173376*n^12-\ 21033288218582573103710208*n^11-55576747262252046136119296*n^10-\ 118697529465947346922289152*n^9-203600243226843972000366848*n^8-\ 277024665807852751135742464*n^7-292777149238353216410433552*n^6-\ 231953324821757706210900464*n^5-128987377247403125517753220*n^4-\ 43137973095293363182106547*n^3-3864366879715031206661523*n^2+ 2711175096050231711437530*n+792490345419546744141600)*x(n+2)+(4*n+15)*( 10303111147028480*n^19+576611488475643904*n^18+15188861899799265280*n^17+ 250272710903883366400*n^16+2891439393774535442432*n^15+24881134691737004933120* n^14+165342758808532645052416*n^13+868111830671527526793216*n^12+ 3653563119674999706157056*n^11+12431167113957899989696512*n^10+ 34327032710530040730867712*n^9+76906998164751357268936192*n^8+ 139180322823943605263223552*n^7+201613120410950857929225344*n^6+ 230283144961655607745023464*n^5+202598927014753259443433366*n^4+ 132337694461799072866880883*n^3+60380898591740320609115931*n^2+ 17157746740315033962082890*n+2283444201992095232818200)*x(n+3)-(4*n+19)*(4*n+15 )*(53519319040*n^12+1498364772352*n^11+19073638858752*n^10+145929218293760*n^9+ 747083375321088*n^8+2695092601595904*n^7+7021984156587776*n^6+13307935687226624 *n^5+18198731009493120*n^4+17504528740711096*n^3+11235522001524711*n^2+ 4318951298386149*n+751602810972618)*(8*n+35)^2*(n+4)^2*(8*n+31)^2*x(n+4) = 0 subject to the initial conditions 832 5102432 52617588544 22792003484576 a(1) = ---, a(2) = -------, a(3) = -----------, a(4) = -------------- 539 563255 749531475 38152077425 b(1) = 3, b(2) = 19, b(3) = 147, b(4) = 1251 Then the limit a(n)/b(n), as n goes to infinity is indeed the constant, 1/2 1/2 2 1/2 4 I 2 dilog(1 - ---- - 1/2 I 2 ) 2 1/2 1/2 2 1/2 + 4 I 2 dilog(1 + ---- - 1/2 I 2 ) 2 1/2 1/2 2 1/2 - 4 I 2 dilog(1 + ---- + 1/2 I 2 ) 2 1/2 1/2 2 1/2 - 4 I 2 dilog(1 - ---- + 1/2 I 2 ) + 32/9 2 1/2 1/2 2 1/2 + 4 2 dilog(1 - ---- - 1/2 I 2 ) 2 1/2 1/2 2 1/2 - 4 2 dilog(1 + ---- - 1/2 I 2 ) 2 1/2 1/2 2 1/2 - 4 2 dilog(1 + ---- + 1/2 I 2 ) 2 1/2 1/2 2 1/2 + 4 2 dilog(1 - ---- + 1/2 I 2 ) 2 The delta such that the error is less than the reciporocal of the denominato\ r to the power delta+1 is approximately, -0.7601399061 This is very disappointing ---------------------------------------- (3/5) (2/5) Diophantine Approximations to the Constant, -10 (-1) dilog(1 + (-1) ) (4/5) (1/5) (2/5) (3/5) + 10 (-1) dilog(1 - (-1) ) + 10 (-1) dilog(1 - (-1) ) 2 (1/5) (4/5) 5 Pi - 10 (-1) dilog(1 + (-1) ) + 50 - -----, 6 in terms of ratios of sequences satisfying the same recurrence By Shalosh B. Ekhad Let a(n), b(n) be the sequences satisfying the linear recurrence 12 -25 (10 n + 11) (5 n + 11) (5 n + 6) (5 n + 8) (n + 2) (62304687500 n 11 10 9 + 2336070312500 n + 39982024609375 n + 413001296875000 n 8 7 6 + 2867418003968750 n + 14095249268568750 n + 50296713200742500 n 5 4 3 + 131257135265660375 n + 248595456167479250 n + 333203612181864370 n 2 + 299975841752924682 n + 162847843506696348 n + 40309644077056224) 3 18 (n + 1) x(n) - (5 n + 11) (n + 2) (5451660156250000 n 17 16 + 248564599609375000 n + 5302859833984375000 n 15 14 + 70317918216308593750 n + 649306364989306640625 n 13 12 + 4432460092067314453125 n + 23176529040625093750000 n 11 10 + 94887214702587246484375 n + 308278276638485479609375 n 9 8 + 800540546087361913984375 n + 1665159894712268203162500 n 7 6 + 2767137031543879103611875 n + 3645659268153714935256250 n 5 4 + 3755532557825705856092500 n + 2957818387758024589040770 n 3 2 + 1718238898615218031588264 n + 693155998432224372527064 n + 173252906357254381956960 n + 20189214528787330937856) x(n + 1) + ( 20 19 18 -54516601562500000 n - 2789381347656250000 n - 67329453686523437500 n 17 16 - 1019257713901367187500 n - 10851264683878662109375 n 15 14 - 86345368526248632812500 n - 532728873840217675781250 n 13 12 - 2609122546733694125000000 n - 10300183761818968917187500 n 11 10 - 33091966626103091420156250 n - 86974833800260010933718750 n 9 8 - 187288561416663498836843750 n - 329763103602058249576015625 n 7 6 - 472040664035219198863976250 n - 543823658141837472062903900 n 5 4 - 496336901162505738859553830 n - 350349947788533572305473428 n 3 2 - 184275928337794174057826448 n - 67922283971196235826851104 n - 15637944001914723358994112 n - 1690796949676722011114496) x(n + 2) + 19 18 (5 n + 16) (18302001953125000 n + 950197827148437500 n 17 16 + 23206510710449218750 n + 354314272190673828125 n 15 14 + 3790521805279687500000 n + 30183236062563525390625 n 13 12 + 185469843065547691406250 n + 899741863476044131250000 n 11 10 + 3495853908350956629453125 n + 10971361549037826442703125 n 9 8 + 27918367469705754577968750 n + 57582633987853052721306875 n 7 6 + 95832616805291232282492125 n + 127517625292966966325179100 n 5 4 + 133628789662088185507179300 n + 107718919208308733531958090 n 3 2 + 64378430323958807445502044 n + 26834364941499640060530048 n + 6954407307257006162740320 n + 842564260306039988927232) x(n + 3) - 2 12 11 (5 n + 21) (5 n + 16) (62304687500 n + 1588414062500 n 10 9 8 + 18397360546875 n + 127957886718750 n + 594986658265625 n 7 6 5 + 1947666861818750 n + 4600093709230000 n + 7894294855461625 n 4 3 2 + 9764013107127375 n + 8483202886044870 n + 4911329700031447 n 2 2 2 + 1700110391569969 n + 265939941490938) (5 n + 18) (n + 4) (10 n + 41) x(n + 4) = 0 and in Maple format -25*(10*n+11)*(5*n+11)*(5*n+6)*(5*n+8)*(n+2)*(62304687500*n^12+2336070312500*n^ 11+39982024609375*n^10+413001296875000*n^9+2867418003968750*n^8+ 14095249268568750*n^7+50296713200742500*n^6+131257135265660375*n^5+ 248595456167479250*n^4+333203612181864370*n^3+299975841752924682*n^2+ 162847843506696348*n+40309644077056224)*(n+1)^3*x(n)-(5*n+11)*(n+2)*( 5451660156250000*n^18+248564599609375000*n^17+5302859833984375000*n^16+ 70317918216308593750*n^15+649306364989306640625*n^14+4432460092067314453125*n^ 13+23176529040625093750000*n^12+94887214702587246484375*n^11+ 308278276638485479609375*n^10+800540546087361913984375*n^9+ 1665159894712268203162500*n^8+2767137031543879103611875*n^7+ 3645659268153714935256250*n^6+3755532557825705856092500*n^5+ 2957818387758024589040770*n^4+1718238898615218031588264*n^3+ 693155998432224372527064*n^2+173252906357254381956960*n+20189214528787330937856 )*x(n+1)+(-54516601562500000*n^20-2789381347656250000*n^19-67329453686523437500 *n^18-1019257713901367187500*n^17-10851264683878662109375*n^16-\ 86345368526248632812500*n^15-532728873840217675781250*n^14-\ 2609122546733694125000000*n^13-10300183761818968917187500*n^12-\ 33091966626103091420156250*n^11-86974833800260010933718750*n^10-\ 187288561416663498836843750*n^9-329763103602058249576015625*n^8-\ 472040664035219198863976250*n^7-543823658141837472062903900*n^6-\ 496336901162505738859553830*n^5-350349947788533572305473428*n^4-\ 184275928337794174057826448*n^3-67922283971196235826851104*n^2-\ 15637944001914723358994112*n-1690796949676722011114496)*x(n+2)+(5*n+16)*( 18302001953125000*n^19+950197827148437500*n^18+23206510710449218750*n^17+ 354314272190673828125*n^16+3790521805279687500000*n^15+30183236062563525390625* n^14+185469843065547691406250*n^13+899741863476044131250000*n^12+ 3495853908350956629453125*n^11+10971361549037826442703125*n^10+ 27918367469705754577968750*n^9+57582633987853052721306875*n^8+ 95832616805291232282492125*n^7+127517625292966966325179100*n^6+ 133628789662088185507179300*n^5+107718919208308733531958090*n^4+ 64378430323958807445502044*n^3+26834364941499640060530048*n^2+ 6954407307257006162740320*n+842564260306039988927232)*x(n+3)-2*(5*n+21)*(5*n+16 )*(62304687500*n^12+1588414062500*n^11+18397360546875*n^10+127957886718750*n^9+ 594986658265625*n^8+1947666861818750*n^7+4600093709230000*n^6+7894294855461625* n^5+9764013107127375*n^4+8483202886044870*n^3+4911329700031447*n^2+ 1700110391569969*n+265939941490938)*(5*n+18)^2*(n+4)^2*(10*n+41)^2*x(n+4) = 0 subject to the initial conditions 75 2552225 699664525 155506018062125 a(1) = --, a(2) = -------, a(3) = ---------, a(4) = --------------- 22 121968 4321152 112855526784 b(1) = 3, b(2) = 19, b(3) = 147, b(4) = 1251 Then the limit a(n)/b(n), as n goes to infinity is indeed the constant, (3/5) (2/5) (4/5) (1/5) -10 (-1) dilog(1 + (-1) ) + 10 (-1) dilog(1 - (-1) ) (2/5) (3/5) (1/5) (4/5) + 10 (-1) dilog(1 - (-1) ) - 10 (-1) dilog(1 + (-1) ) 2 5 Pi + 50 - ----- 6 The delta such that the error is less than the reciporocal of the denominato\ r to the power delta+1 is approximately, -0.7580414342 This is very disappointing ---------------------------------------- (3/5) (1/5) Diophantine Approximations to the Constant, 10 (-1) dilog(1 - (-1) ) (4/5) (3/5) + 25/2 - 10 (-1) dilog(1 - (-1) ) (1/5) (2/5) (2/5) (4/5) + 10 (-1) dilog(1 + (-1) ) - 10 (-1) dilog(1 + (-1) ) 2 5 Pi + -----, in terms of ratios of sequences satisfying the same recurrence 6 By Shalosh B. Ekhad Let a(n), b(n) be the sequences satisfying the linear recurrence 12 -25 (10 n + 17) (5 n + 12) (5 n + 7) (5 n + 6) (n + 2) (62304687500 n 11 10 9 + 2392765625000 n + 41950979296875 n + 443955087890625 n 8 7 6 + 3158162188578125 n + 15908001679918750 n + 58173289168935625 n 5 4 3 + 155592186758033625 n + 302048440866730375 n + 414999349683004440 n 2 + 383011874149700988 n + 213171207951742632 n + 54101225727493776) 3 18 (n + 1) x(n) - (5 n + 12) (n + 2) (5451660156250000 n 17 16 + 255238818359375000 n + 5592387333984375000 n 15 14 + 76172908174316406250 n + 722593346609423828125 n 13 12 + 5068209566008359375000 n + 27231476649685072265625 n 11 10 + 114572828731480325390625 n + 382555842973473394453125 n 9 8 + 1021003043792111185078125 n + 2182681349447515966540625 n 7 6 + 3727673675245229025223125 n + 5046842030116509652996250 n 5 4 + 5341891552489338472621375 n + 4322122772241312953852130 n 3 2 + 2578761923037611464859012 n + 1068165214400869259008752 n + 274045711704493312664640 n + 32766496019891113565088) x(n + 1) + ( 20 19 18 -54516601562500000 n - 2844441406250000000 n - 69971179370117187500 n 17 16 - 1078747073369140625000 n - 11686626182775146484375 n 15 14 - 94541267921296582031250 n - 592384783197280322265625 n 13 12 - 2942941779908720802734375 n - 11768421204116250126171875 n 11 10 - 38237407212853848761796875 n - 101450533291215883446984375 n 9 8 - 220063255684923996423515625 n - 389360954056957452511056250 n 7 6 - 558489788461955480550104375 n - 642618916728417260522159600 n 5 4 - 583540678922017207012180340 n - 407994769716040666342531968 n 3 2 - 211445600163117314514182208 n - 76317576049346966004662784 n - 17079057357673135802487552 n - 1779036231895502546912256) x(n + 2) + 19 18 (5 n + 17) (18302001953125000 n + 977132348632812500 n 17 16 + 24547140388183593750 n + 385612346054443359375 n 15 14 + 4245799132865136718750 n + 34806391285812255859375 n 13 12 + 220263094111120775390625 n + 1100809587305758012890625 n 11 10 + 4407906735246246459140625 n + 14262385828898375043140625 n 9 8 + 37432679574308341310890625 n + 79665047072416683519417500 n 7 6 + 136867749703196987743377125 n + 188094530354158973841565200 n 5 4 + 203677108913076296924174700 n + 169746513578811973095130800 n 3 2 + 104944582049084293189387392 n + 45277340291493210766920384 n + 12153340429934736194668800 n + 1526086611514427656421376) x(n + 3) - 2 12 11 (5 n + 22) (5 n + 17) (62304687500 n + 1645109375000 n 10 9 8 + 19742666796875 n + 142340373046875 n + 686394958109375 n 7 6 5 + 2331237163481250 n + 5715408771723125 n + 10186211159994875 n 4 3 2 + 13090731756096625 n + 11823855818550440 n + 7120367160958668 n 2 2 2 + 2565317025464976 n + 417911459208192) (10 n + 37) (n + 4) (5 n + 21) x(n + 4) = 0 and in Maple format -25*(10*n+17)*(5*n+12)*(5*n+7)*(5*n+6)*(n+2)*(62304687500*n^12+2392765625000*n^ 11+41950979296875*n^10+443955087890625*n^9+3158162188578125*n^8+ 15908001679918750*n^7+58173289168935625*n^6+155592186758033625*n^5+ 302048440866730375*n^4+414999349683004440*n^3+383011874149700988*n^2+ 213171207951742632*n+54101225727493776)*(n+1)^3*x(n)-(5*n+12)*(n+2)*( 5451660156250000*n^18+255238818359375000*n^17+5592387333984375000*n^16+ 76172908174316406250*n^15+722593346609423828125*n^14+5068209566008359375000*n^ 13+27231476649685072265625*n^12+114572828731480325390625*n^11+ 382555842973473394453125*n^10+1021003043792111185078125*n^9+ 2182681349447515966540625*n^8+3727673675245229025223125*n^7+ 5046842030116509652996250*n^6+5341891552489338472621375*n^5+ 4322122772241312953852130*n^4+2578761923037611464859012*n^3+ 1068165214400869259008752*n^2+274045711704493312664640*n+ 32766496019891113565088)*x(n+1)+(-54516601562500000*n^20-2844441406250000000*n^ 19-69971179370117187500*n^18-1078747073369140625000*n^17-\ 11686626182775146484375*n^16-94541267921296582031250*n^15-\ 592384783197280322265625*n^14-2942941779908720802734375*n^13-\ 11768421204116250126171875*n^12-38237407212853848761796875*n^11-\ 101450533291215883446984375*n^10-220063255684923996423515625*n^9-\ 389360954056957452511056250*n^8-558489788461955480550104375*n^7-\ 642618916728417260522159600*n^6-583540678922017207012180340*n^5-\ 407994769716040666342531968*n^4-211445600163117314514182208*n^3-\ 76317576049346966004662784*n^2-17079057357673135802487552*n-\ 1779036231895502546912256)*x(n+2)+(5*n+17)*(18302001953125000*n^19+ 977132348632812500*n^18+24547140388183593750*n^17+385612346054443359375*n^16+ 4245799132865136718750*n^15+34806391285812255859375*n^14+ 220263094111120775390625*n^13+1100809587305758012890625*n^12+ 4407906735246246459140625*n^11+14262385828898375043140625*n^10+ 37432679574308341310890625*n^9+79665047072416683519417500*n^8+ 136867749703196987743377125*n^7+188094530354158973841565200*n^6+ 203677108913076296924174700*n^5+169746513578811973095130800*n^4+ 104944582049084293189387392*n^3+45277340291493210766920384*n^2+ 12153340429934736194668800*n+1526086611514427656421376)*x(n+3)-2*(5*n+22)*(5*n+ 17)*(62304687500*n^12+1645109375000*n^11+19742666796875*n^10+142340373046875*n^ 9+686394958109375*n^8+2331237163481250*n^7+5715408771723125*n^6+ 10186211159994875*n^5+13090731756096625*n^4+11823855818550440*n^3+ 7120367160958668*n^2+2565317025464976*n+417911459208192)*(10*n+37)^2*(n+4)^2*(5 *n+21)^2*x(n+4) = 0 subject to the initial conditions 725 9814925 3320994025 35785659087125 a(1) = ---, a(2) = -------, a(3) = ----------, a(4) = -------------- 294 659736 28839888 36517707072 b(1) = 3, b(2) = 19, b(3) = 147, b(4) = 1251 Then the limit a(n)/b(n), as n goes to infinity is indeed the constant, (3/5) (1/5) 10 (-1) dilog(1 - (-1) ) + 25/2 (4/5) (3/5) (1/5) (2/5) - 10 (-1) dilog(1 - (-1) ) + 10 (-1) dilog(1 + (-1) ) 2 (2/5) (4/5) 5 Pi - 10 (-1) dilog(1 + (-1) ) + ----- 6 The delta such that the error is less than the reciporocal of the denominato\ r to the power delta+1 is approximately, -0.7586705462 This is very disappointing ---------------------------------------- Diophantine Approximations to the Constant, 50/9 (3/5) (4/5) (2/5) (1/5) - 10 (-1) dilog(1 + (-1) ) + 10 (-1) dilog(1 - (-1) ) (1/5) (3/5) (4/5) (2/5) - 10 (-1) dilog(1 - (-1) ) + 10 (-1) dilog(1 + (-1) ) 2 5 Pi - -----, in terms of ratios of sequences satisfying the same recurrence 6 By Shalosh B. Ekhad Let a(n), b(n) be the sequences satisfying the linear recurrence 12 -25 (5 n + 9) (5 n + 13) (5 n + 8) (10 n + 13) (n + 2) (62304687500 n 11 10 9 + 2449460937500 n + 43964686328125 n + 476335478515625 n 8 7 6 + 3469288047093750 n + 17892595585518750 n + 66996611295623125 n 5 4 3 + 183488327144832625 n + 364761657286391375 n + 513231525696330085 n 2 + 485103309410175228 n + 276523017212503692 n + 71881602733858176) 3 18 (n + 1) x(n) - (5 n + 13) (n + 2) (5451660156250000 n 17 16 + 261913037109375000 n + 5889128032226562500 n 15 14 + 82323506107910156250 n + 801502316538378906250 n 13 12 + 5769799842489101562500 n + 31817671960769623046875 n 11 10 + 137388655402733533984375 n + 470760058678386374921875 n 9 8 + 1289169717585881104218750 n + 2827287982565867302118750 n 7 6 + 4952267887437989262776875 n + 6874313993021045229398125 n 5 4 + 7457049281916693828351500 n + 6180259230099684330822105 n 3 2 + 3774714521207754033660918 n + 1599349179990064012234848 n + 419336862992060787694440 n + 51183702655015470866352) x(n + 1) + ( 20 19 18 -54516601562500000 n - 2899501464843750000 n - 72643721972656250000 n 17 16 - 1139488054462890625000 n - 12544919083911621093750 n 15 14 - 102983351771966357421875 n - 653691924734959423828125 n 13 12 - 3283071138162950691406250 n - 13239215662447319735546875 n 11 10 - 43246622022564874322578125 n - 114920930671798302421671875 n 9 8 - 248497630076844826082546875 n - 435662804002832818677153125 n 7 6 - 614416545785820369910270000 n - 687992456436876961488365100 n 5 4 - 599511049616404292671985735 n - 394337334819928862303094118 n 3 2 - 186677136112614677594749548 n - 58708247389062930903647304 n - 10517752840698614229623712 n - 726593980933989569097216) x(n + 2) + 19 18 (5 n + 18) (18302001953125000 n + 1004066870117187500 n 17 16 + 25924173762207031250 n + 418640320044189453125 n 15 14 + 4739506266361474609375 n + 39959497192740917968750 n 13 12 + 260136127754072562500000 n + 1337786288322166604296875 n 11 10 + 5513759184258500512265625 n + 18368843554941337221000000 n 9 8 + 49654308368191431011421875 n + 108878452435013176487766875 n 7 6 + 192799368200307307049495250 n + 273202700268210463550594050 n 5 4 + 305170500407631278839409575 n + 262479153924167631595448365 n 3 2 + 167558636800729455266539458 n + 74686038582367679241441276 n + 20723708131448823539759520 n + 2691856951546158735519024) x(n + 3) - 2 12 11 (5 n + 23) (5 n + 18) (62304687500 n + 1701804687500 n 10 9 8 + 21132725390625 n + 157701935546875 n + 787359390843750 n 7 6 5 + 2769582872831250 n + 7034830027335625 n + 12994511008643875 n 4 3 2 + 17315341346856375 n + 16223528801909585 n + 10139857464300723 n 2 2 2 + 3793747865305491 n + 642245185519002) (5 n + 19) (n + 4) (10 n + 43) x(n + 4) = 0 and in Maple format -25*(5*n+9)*(5*n+13)*(5*n+8)*(10*n+13)*(n+2)*(62304687500*n^12+2449460937500*n^ 11+43964686328125*n^10+476335478515625*n^9+3469288047093750*n^8+ 17892595585518750*n^7+66996611295623125*n^6+183488327144832625*n^5+ 364761657286391375*n^4+513231525696330085*n^3+485103309410175228*n^2+ 276523017212503692*n+71881602733858176)*(n+1)^3*x(n)-(5*n+13)*(n+2)*( 5451660156250000*n^18+261913037109375000*n^17+5889128032226562500*n^16+ 82323506107910156250*n^15+801502316538378906250*n^14+5769799842489101562500*n^ 13+31817671960769623046875*n^12+137388655402733533984375*n^11+ 470760058678386374921875*n^10+1289169717585881104218750*n^9+ 2827287982565867302118750*n^8+4952267887437989262776875*n^7+ 6874313993021045229398125*n^6+7457049281916693828351500*n^5+ 6180259230099684330822105*n^4+3774714521207754033660918*n^3+ 1599349179990064012234848*n^2+419336862992060787694440*n+ 51183702655015470866352)*x(n+1)+(-54516601562500000*n^20-2899501464843750000*n^ 19-72643721972656250000*n^18-1139488054462890625000*n^17-\ 12544919083911621093750*n^16-102983351771966357421875*n^15-\ 653691924734959423828125*n^14-3283071138162950691406250*n^13-\ 13239215662447319735546875*n^12-43246622022564874322578125*n^11-\ 114920930671798302421671875*n^10-248497630076844826082546875*n^9-\ 435662804002832818677153125*n^8-614416545785820369910270000*n^7-\ 687992456436876961488365100*n^6-599511049616404292671985735*n^5-\ 394337334819928862303094118*n^4-186677136112614677594749548*n^3-\ 58708247389062930903647304*n^2-10517752840698614229623712*n-\ 726593980933989569097216)*x(n+2)+(5*n+18)*(18302001953125000*n^19+ 1004066870117187500*n^18+25924173762207031250*n^17+418640320044189453125*n^16+ 4739506266361474609375*n^15+39959497192740917968750*n^14+ 260136127754072562500000*n^13+1337786288322166604296875*n^12+ 5513759184258500512265625*n^11+18368843554941337221000000*n^10+ 49654308368191431011421875*n^9+108878452435013176487766875*n^8+ 192799368200307307049495250*n^7+273202700268210463550594050*n^6+ 305170500407631278839409575*n^5+262479153924167631595448365*n^4+ 167558636800729455266539458*n^3+74686038582367679241441276*n^2+ 20723708131448823539759520*n+2691856951546158735519024)*x(n+3)-2*(5*n+23)*(5*n+ 18)*(62304687500*n^12+1701804687500*n^11+21132725390625*n^10+157701935546875*n^ 9+787359390843750*n^8+2769582872831250*n^7+7034830027335625*n^6+ 12994511008643875*n^5+17315341346856375*n^4+16223528801909585*n^3+ 10139857464300723*n^2+3793747865305491*n+642245185519002)*(5*n+19)^2*(n+4)^2*( 10*n+43)^2*x(n+4) = 0 subject to the initial conditions 775 4126475 7382539225 1180546530286625 a(1) = ---, a(2) = -------, a(3) = ----------, a(4) = ---------------- 416 373152 86198112 1619748722592 b(1) = 3, b(2) = 19, b(3) = 147, b(4) = 1251 Then the limit a(n)/b(n), as n goes to infinity is indeed the constant, 50/9 (3/5) (4/5) (2/5) (1/5) - 10 (-1) dilog(1 + (-1) ) + 10 (-1) dilog(1 - (-1) ) (1/5) (3/5) (4/5) (2/5) - 10 (-1) dilog(1 - (-1) ) + 10 (-1) dilog(1 + (-1) ) 2 5 Pi - ----- 6 The delta such that the error is less than the reciporocal of the denominato\ r to the power delta+1 is approximately, -0.7574323158 This is very disappointing ---------------------------------------- Diophantine Approximations to the Constant, 25/8 (3/5) (3/5) (1/5) (1/5) + 10 (-1) dilog(1 - (-1) ) + 10 (-1) dilog(1 - (-1) ) (2/5) (2/5) (4/5) (4/5) - 10 (-1) dilog(1 + (-1) ) - 10 (-1) dilog(1 + (-1) ) 2 5 Pi + -----, in terms of ratios of sequences satisfying the same recurrence 6 By Shalosh B. Ekhad Let a(n), b(n) be the sequences satisfying the linear recurrence 12 -25 (5 n + 7) (5 n + 14) (5 n + 9) (10 n + 19) (n + 2) (62304687500 n 11 10 9 + 2506156250000 n + 46023145703125 n + 510172960937500 n 8 7 6 + 3801678853812500 n + 20060359167806250 n + 76851098740101875 n 5 4 3 + 215348360695195250 n + 438011167952780000 n + 630583838810217280 n 2 + 609862963805822352 n + 355727983756487328 n + 94627983606695424) 3 18 (n + 1) x(n) - (5 n + 14) (n + 2) (5451660156250000 n 17 16 + 268587255859375000 n + 6193081928710937500 n 15 14 + 88776481710449218750 n + 886292930366015625000 n 13 12 + 6541826009906220703125 n + 36984855594655064453125 n 11 10 + 163702765379059694921875 n + 574861648480487163203125 n 9 8 + 1612920892253739274140625 n + 3622919593003824178021875 n 7 6 + 6496561658334131748856875 n + 9226878867430218533644375 n 5 4 + 10233717053549201253979750 n + 8664344691669797975363320 n 3 2 + 5400176676422844380870656 n + 2331758594727340343244336 n + 622017702171805846966560 n + 77086182715777241685504) x(n + 1) + ( 20 19 18 -54516601562500000 n - 2954561523437500000 n - 75347081494140625000 n 17 16 - 1201451612246093750000 n - 13424430700354492187500 n 15 14 - 111626491073941796875000 n - 715934167890105322265625 n 13 12 - 3621780885768434966796875 n - 14652023960485431036328125 n 11 10 - 47762048093239285642734375 n - 125755726387400678350359375 n 9 8 - 266772178370353967531703125 n - 452292941299271654813271875 n 7 6 - 603464700563632797965248125 n - 616585160811710123525267900 n 5 4 - 458599360329944662025967220 n - 221430127454641262606526728 n 3 2 - 43712684042690099101715328 n + 19419152958941816692580256 n + 15144500857765540900560768 n + 3125425571363373907958784) x(n + 2) + 19 18 (5 n + 19) (18302001953125000 n + 1031001391601562500 n 17 16 + 27337610832519531250 n + 453443233927490234375 n 15 14 + 5273677701770654296875 n + 45685355955294824218750 n 13 12 + 305645543029234546875000 n + 1615678399763339103125000 n 11 10 + 6846431537957384495468750 n + 23455893788993916267531250 n 9 8 + 65222133073063344846281250 n + 147153970457695792170805625 n 7 6 + 268202588824175329933778375 n + 391307204685333013520693900 n 5 4 + 450208715922623166624768300 n + 399010963335530407278760920 n 3 2 + 262588491746287483008185856 n + 120723217255428268566616992 n + 34571046933511596776741760 n + 4637396916135832116538368) x(n + 3) - 2 12 11 (5 n + 24) (5 n + 19) (62304687500 n + 1758500000000 n 10 9 8 + 22567536328125 n + 174073066406250 n + 898488801078125 n 7 6 5 + 3268063696681250 n + 8585649694864375 n + 16407754200952750 n 4 3 2 + 22627671249331875 n + 21950914193106030 n + 14211883102760512 n 2 2 2 + 5511340397789464 n + 967756862709168) (5 n + 22) (10 n + 39) (n + 4) x(n + 4) = 0 and in Maple format -25*(5*n+7)*(5*n+14)*(5*n+9)*(10*n+19)*(n+2)*(62304687500*n^12+2506156250000*n^ 11+46023145703125*n^10+510172960937500*n^9+3801678853812500*n^8+ 20060359167806250*n^7+76851098740101875*n^6+215348360695195250*n^5+ 438011167952780000*n^4+630583838810217280*n^3+609862963805822352*n^2+ 355727983756487328*n+94627983606695424)*(n+1)^3*x(n)-(5*n+14)*(n+2)*( 5451660156250000*n^18+268587255859375000*n^17+6193081928710937500*n^16+ 88776481710449218750*n^15+886292930366015625000*n^14+6541826009906220703125*n^ 13+36984855594655064453125*n^12+163702765379059694921875*n^11+ 574861648480487163203125*n^10+1612920892253739274140625*n^9+ 3622919593003824178021875*n^8+6496561658334131748856875*n^7+ 9226878867430218533644375*n^6+10233717053549201253979750*n^5+ 8664344691669797975363320*n^4+5400176676422844380870656*n^3+ 2331758594727340343244336*n^2+622017702171805846966560*n+ 77086182715777241685504)*x(n+1)+(-54516601562500000*n^20-2954561523437500000*n^ 19-75347081494140625000*n^18-1201451612246093750000*n^17-\ 13424430700354492187500*n^16-111626491073941796875000*n^15-\ 715934167890105322265625*n^14-3621780885768434966796875*n^13-\ 14652023960485431036328125*n^12-47762048093239285642734375*n^11-\ 125755726387400678350359375*n^10-266772178370353967531703125*n^9-\ 452292941299271654813271875*n^8-603464700563632797965248125*n^7-\ 616585160811710123525267900*n^6-458599360329944662025967220*n^5-\ 221430127454641262606526728*n^4-43712684042690099101715328*n^3+ 19419152958941816692580256*n^2+15144500857765540900560768*n+ 3125425571363373907958784)*x(n+2)+(5*n+19)*(18302001953125000*n^19+ 1031001391601562500*n^18+27337610832519531250*n^17+453443233927490234375*n^16+ 5273677701770654296875*n^15+45685355955294824218750*n^14+ 305645543029234546875000*n^13+1615678399763339103125000*n^12+ 6846431537957384495468750*n^11+23455893788993916267531250*n^10+ 65222133073063344846281250*n^9+147153970457695792170805625*n^8+ 268202588824175329933778375*n^7+391307204685333013520693900*n^6+ 450208715922623166624768300*n^5+399010963335530407278760920*n^4+ 262588491746287483008185856*n^3+120723217255428268566616992*n^2+ 34571046933511596776741760*n+4637396916135832116538368)*x(n+3)-2*(5*n+24)*(5*n+ 19)*(62304687500*n^12+1758500000000*n^11+22567536328125*n^10+174073066406250*n^ 9+898488801078125*n^8+3268063696681250*n^7+8585649694864375*n^6+ 16407754200952750*n^5+22627671249331875*n^4+21950914193106030*n^3+ 14211883102760512*n^2+5511340397789464*n+967756862709168)*(5*n+22)^2*(10*n+39)^ 2*(n+4)^2*x(n+4) = 0 subject to the initial conditions 275 5132725 17745683075 201551931590125 a(1) = ---, a(2) = -------, a(3) = -----------, a(4) = --------------- 189 603288 269095176 359152361568 b(1) = 3, b(2) = 19, b(3) = 147, b(4) = 1251 Then the limit a(n)/b(n), as n goes to infinity is indeed the constant, 25/8 (3/5) (3/5) (1/5) (1/5) + 10 (-1) dilog(1 - (-1) ) + 10 (-1) dilog(1 - (-1) ) (2/5) (2/5) (4/5) (4/5) - 10 (-1) dilog(1 + (-1) ) - 10 (-1) dilog(1 + (-1) ) 2 5 Pi + ----- 6 The delta such that the error is less than the reciporocal of the denominato\ r to the power delta+1 is approximately, -0.7540561857 This is very disappointing ----------------------------------------- This took, 64.866, seconds.