Rational generating functions for the Certain Stanely-Stern Sums By Shalosh B. Ekhad Theorem Number, 1 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- \ n 1 ) B(n) t = - ------- / 3 t - 1 ----- n = 0 and in Maple notation -1/(3*t-1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, 1162261467, 3486784401, 10460353203, 31381059609, 94143178827, 282429536481, 847288609443, 2541865828329, 7625597484987, 22876792454961, 68630377364883, 205891132094649 ----------------------------- This took, 0.007, seconds. Theorem Number, 2 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 2 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- \ n 2 t - 1 ) B(n) t = - -------------- / 2 ----- 2 t - 5 t + 1 n = 0 and in Maple notation -(2*t-1)/(2*t^2-5*t+1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 13, 59, 269, 1227, 5597, 25531, 116461, 531243, 2423293, 11053979, 50423309, 230008587, 1049196317, 4785964411, 21831429421, 99585218283, 454263232573, 2072145726299, 9452202166349, 43116719379147, 196679192563037, 897162524056891, 4092454235158381, 18667946127678123, 85154822168073853, 388438218585013019, 1771881448588917389, 8082530805774560907, 36868891131694969757 ----------------------------- This took, 0.003, seconds. Theorem Number, 3 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 3 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- \ n 4 t - 1 ) B(n) t = ------- / 7 t - 1 ----- n = 0 and in Maple notation (4*t-1)/(7*t-1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 21, 147, 1029, 7203, 50421, 352947, 2470629, 17294403, 121060821, 847425747, 5931980229, 41523861603, 290667031221, 2034669218547, 14242684529829, 99698791708803, 697891541961621, 4885240793731347, 34196685556119429, 239376798892836003, 1675637592249852021, 11729463145748964147, 82106242020242749029, 574743694141699243203, 4023205858991894702421, 28162441012943262916947, 197137087090602840418629, 1379959609634219882930403, 9659717267439539180512821 ----------------------------- This took, 0.013, seconds. Theorem Number, 4 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 4 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- 2 \ n 2 t + 7 t - 1 ) B(n) t = - ------------------------- / 2 ----- (2 t - 11 t + 1) (t + 1) n = 0 and in Maple notation -(2*t^2+7*t-1)/(2*t^2-11*t+1)/(t+1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 37, 395, 4277, 46251, 500213, 5409835, 58507765, 632765739, 6843407605, 74011952171, 800444658677, 8656867341099, 93624651434741, 1012557431099947, 10950882439229941, 118434591969329451, 1280878746784164085, 13852797030687146027, 149819009843990278133, 1620303514222518767403, 17523700636759725885173, 189520099975911947202091, 2049673698461511967452661, 22167370483124807747575083, 239741727917449861288420597, 2592824266125698858677476395, 28041583471547787722875399157, 303271769654774267234274437931, 3279906299259421364131268018933 ----------------------------- This took, 0.019, seconds. Theorem Number, 5 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 5 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- 2 \ n 20 t + 11 t - 1 ) B(n) t = ---------------- / 2 ----- 47 t + 14 t - 1 n = 0 and in Maple notation (20*t^2+11*t-1)/(47*t^2+14*t-1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 69, 1107, 18741, 314403, 5282469, 88731507, 1490517141, 25037620803, 420580996869, 7064902133907, 118675936727541, 1993513514479203, 33486958228903269, 562512550385168307, 9449062742150809941, 158724968258214249603, 2666255504496087561669, 44787650571081295594707, 752341116706454253724341, 12637795210731180445092003, 212289165435439876156332069, 3566024691000523747107973107, 59901936449473006638859230741, 1006230270769646709058103966403, 16902614803900285238839839374469, 283929429980777388669488637663507, 4769434915514196847598313377889141, 80116772026295293133842353260632803, 1345798249397301355710913674409648869 ----------------------------- This took, 0.053, seconds. Theorem Number, 6 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 6 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- 3 2 \ n 4 t + 88 t + 17 t - 1 ) B(n) t = - -------------------------------- / 4 3 2 ----- 4 t - 40 t - 161 t - 20 t + 1 n = 0 and in Maple notation -(4*t^3+88*t^2+17*t-1)/(4*t^4-40*t^3-161*t^2-20*t+1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 133, 3179, 85109, 2219307, 58215317, 1525006411, 39961226101, 1047070289643, 27435930590293, 718889277456779, 18836693340854069, 493568289430075947, 12932719243854934037, 338869471651865243851, 8879224616101788878581, 232657811754582558889323, 6096214446273138377255893, 159736010124708023620466699, 4185481523915854646347591349, 109670046054998991934701643947, 2873628501997635665147304730517, 75296227771723686358192481072971, 1972948804172197365511368358465141, 51696175214587146244534510004696043, 1354568616360327657649687615830805333, 35493073304010884698238830391786483339, 930006968527597243906695648911243306549, 24368500132451252072271374037895703377707, 638515429239663191791704434697832376020757 ----------------------------- This took, 0.161, seconds. Theorem Number, 7 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 7 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- 3 2 \ n 84 t + 311 t + 26 t - 1 ) B(n) t = -------------------------- / 3 2 ----- 327 t + 485 t + 29 t - 1 n = 0 and in Maple notation (84*t^3+311*t^2+26*t-1)/(327*t^3+485*t^2+29*t-1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 261, 9267, 396309, 16072803, 661351461, 27104094867, 1112030016309, 45610618411203, 1870905530856261, 76741044139598067, 3147794134734093909, 129117222423583783203, 5296173927063613826661, 217240225442340984534867, 8910832224186253154549109, 365507492715086520702652803, 14992508471187487382276150661, 614967721768563001408332537267, 25224951489932091713515344117909, 1034685488535862023748990561260003, 42441075085175383271236192913412261, 1740862198547186990374429534226613267, 71407267328929710489173929969108752309, 2929007150387199644847336528286747060803, 120142993954684629433674553813025512171461, 4928065469040206096657480151896710739466867, 202141036008364636362264162049071751136580309, 8291488555750256285209350152189844834082327203, 340103047989190428300376269016975523901434604261 ----------------------------- This took, 0.480, seconds. Theorem Number, 8 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 8 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- 4 3 2 \ n 4 t + 428 t + 969 t + 40 t - 1 ) B(n) t = - ------------------------------------------- / 4 3 2 ----- (t + 1) (4 t - 88 t - 1313 t - 44 t + 1) n = 0 and in Maple notation -(4*t^4+428*t^3+969*t^2+40*t-1)/(t+1)/(4*t^4-88*t^3-1313*t^2-44*t+1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 517, 27275, 1878677, 118519851, 7683973973, 493876634155, 31830051961045, 2049658022562219, 132021241625498965, 8502934684212183851, 547653258945393861077, 35272706304598880944299, 2271815536564897071325781, 146321106461842259629238827, 9424124291372538198284912341, 606980860281183060164479614891, 39093900217050693160844280672085, 2517926216752638793065520639254827, 162172421141311245634213129716120533, 10445061488109568879466319945631575723, 672736515566836060225257039364187327573, 43329033520184218967711226313970962925611, 2790699036548630458471466949802597044011221, 179740937653265548812373690695742410216345003, 11576599355735749883642396182590947956294056277, 745615630990192859214381706164188867347020750635, 48022968757366866576428916257879650951593871234517, 3093021970593819486438920776909589845147281591475371, 199212692553680467246775012251175653142457855605667413 ----------------------------- This took, 1.583, seconds. Theorem Number, 9 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 9 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- 3 2 \ n 900 t + 2819 t + 62 t - 1 ) B(n) t = ---------------------------- / 3 2 ----- 3843 t + 3653 t + 65 t - 1 n = 0 and in Maple notation (900*t^3+2819*t^2+62*t-1)/(3843*t^3+3653*t^2+65*t-1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 1029, 80787, 9021621, 885474723, 90822302949, 9172758944307, 931406083413141, 94398513955640643, 9573580742447817669, 970699913317620118227, 98430558306938712620661, 9980744344093509575528163, 1012045611628204853726951589, 102620892480380471444257432947, 10405716631016914331840042026581, 1055134992532416485008347809239683, 106990229457513961731514562438495109, 10848762211472322934061202387162894867, 1100059735730301569471087932384318901301, 111545574632783223848680498388737920237603, 11310672358932389320477713813041748766870629, 1146897217028573971481758649385926230012254387, 116294874877351112263269879924434633023648051221, 11792249314708580187094002395478559105282972432323, 1195728909388062135031239425559656170780716632820549, 121246387061008142524912699560610429807055158889453907, 12294330479076545317047445344214926031962145650565386741, 1246639719272616513036515091981305850827022256729854630243, 126408720858262147682271038325613430980262677724568080095269 ----------------------------- This took, 5.431, seconds. Theorem Number, 10 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 10 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- 4 3 2 \ n 4 t + 1852 t + 7945 t + 96 t - 1 ) B(n) t = - --------------------------------------------- / 4 3 2 ----- (t + 1) (4 t - 200 t - 9601 t - 100 t + 1) n = 0 and in Maple notation -(4*t^4+1852*t^3+7945*t^2+96*t-1)/(t+1)/(4*t^4-200*t^3-9601*t^2-100*t+1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 2053, 240299, 43735349, 6681062187, 1088057350037, 172959359175691, 27743710572764341, 4435171449468008043, 709919097675516974293, 113579538904171158821579, 18174774070514805662890229, 2908096526149277167657835307, 465328371534044809464889031957, 74457106401459638377619722743691, 11913909881850461626808680785682741, 1906346720787380855109179343805507563, 305035010414351173381404364688883357013, 48808718391862705414530475574443259393099, 7809894195862974105857086191456333008767349, 1249662924243267232355768575572083435759256747, 199958847122345468509588954650878026689446178197, 31995460231498454576176979318271764704497957751051, 5119600815716756170772836215717026055556988348248501, 819188482025065051583791788374285691800587873391309163, 131078533926413992355502540702035572088890225304430265813, 20973905800745351221046575292920300267990519282550804557259, 3356039401520037612853129497838603548694742876749640306629109, 537000622174991233056792013344161411359726274149443697705086507, 85925590768339017791270337536212995058902163522913275297645108757 ----------------------------- This took, 19.782, seconds. Theorem Number, 11 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 11 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- 4 3 2 \ n 18348 t + 86213 t - 21249 t - 153 t + 1 ) B(n) t = ------------------------------------------- / 4 3 2 ----- 107529 t + 83828 t - 24882 t - 156 t + 1 n = 0 and in Maple notation (18348*t^4+86213*t^3-21249*t^2-153*t+1)/(107529*t^4+83828*t^3-24882*t^2-156*t+1 ) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 4101, 716787, 213519189, 50799986403, 13177054542501, 3301649808344787, 838647420463231989, 211870579523572478403, 53640847906011617586501, 13569078873978509433342387, 3433617016079251515796618389, 868750687820885776080626340003, 219817129203559761272623483312101, 55618434453408471107760351188797587, 14072770737511924274122503457815598389, 3560729874921983065171825942670093923203, 900946523139154603061929759443910780103301, 227960065537494410557725527996102612495766387, 57679119517677978964299622698078931492996459989, 14594137568597271103945577419352884991623139388003, 3692650998287474053201667627586689498019378829432101, 934325249165868161050741836812027592465576087131912787, 236405681467121374078178018414249697141845758681544259189, 59816050317713009868564003698300160706246169821268271662403, 15134830331771872319307181722303580616301571821613790646978501, 3829458613236003786177814756553862890708865524474718776823188787, 968940711647408593407245161930597132118031562427087923612974259989, 245164185714407600904185592785575429983147005611548716690200067642403, 62032152468857313539165689956351070118609361140810328484091222868390501 ----------------------------- This took, 70.433, seconds. Theorem Number, 12 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 12 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- \ n ) B(n) t = / ----- n = 0 6 5 4 3 2 8 t + 15404 t - 9582 t - 667319 t + 56447 t + 243 t - 1 - ---------------------------------------------------------------------- 6 5 4 3 2 (t + 1) (8 t - 988 t - 127318 t + 797003 t - 63659 t - 247 t + 1) and in Maple notation -(8*t^6+15404*t^5-9582*t^4-667319*t^3+56447*t^2+243*t-1)/(t+1)/(8*t^6-988*t^5-\ 127318*t^4+797003*t^3-63659*t^2-247*t+1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 8197, 2142155, 1047947477, 388678519851, 161008621810133, 63677070942973675, 25668239857061509525, 10265399036051558656299, 4118837726072368390909525, 1650388398801475297223341931, 661668739571948876787281152277, 265212834724815816262293335440299, 106313900376753844931373966816075221, 42615575396579675931243711243354377707, 17082592526095906198196179273198418619541, 6847566537595987482079649809972730910514731, 2744858486902341164192645471370554102932972885, 1100279832825489572750346123564799443482131922027, 441048709016067021288698231175447910517785132070933, 176794956391523462492251542064993337219206231133878443, 70868497145016786588529597021212909376968182996800885973, 28407731867654827002201530755103711070773357294267999249131, 11387277474963041536390976671133008572361039978340116775512981, 4564605444064845255913709831292786988469428345987775467796007723, 1829728216950961937062951816437972416584900737780944956926854556757, 733449010126856200504021454486883674703156298603322997539043564634475, 294004019664938583094272893634465528072538414718917683962251006762125077, 117851905682497131706822758933383567825796969196130275948847089171336746411 , 47241094491855846863501403370541591990428173891511363875576678981162634\ 177493 ----------------------------- This took, 270.925, seconds. Theorem Number, 13 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 13 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- 5 4 3 2 \ n 289380 t + 1565891 t - 4165856 t + 148038 t + 388 t - 1 ) B(n) t = ------------------------------------------------------------ / 5 4 3 2 ----- 2493147 t - 2327461 t - 4653650 t + 163254 t + 391 t - 1 n = 0 and in Maple notation (289380*t^5+1565891*t^4-4165856*t^3+148038*t^2+388*t-1)/(2493147*t^5-2327461*t^ 4-4653650*t^3+163254*t^2+391*t-1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 16389, 6410067, 5164051701, 2989339844643, 1982015630077029, 1238943230838344307, 794075438414668269141, 503515382893346242274883, 320740292556839044295721669, 203912126894021324601258412947, 129746747884288549608884506323381, 82526665791586458931717457637364323, 52499920979107397664158339242362159909, 33396007648808854125676542089280613585587, 21244309399833774652133068646558828024928021, 13514048796650848715136470505328610551601554563, 8596676249488592854689267822917193897392125865349, 5468581758344603201908732982683524658764590483791827, 3478718236723772602810424098008667917415608612860972661, 2212909354051012294171433797869111585299817928695745879203, 1407693384216138121083540369076634153787220534818246296879589, 895472973314434124938482699630997254747005959032943562627265267, 569635319750237904030614671352173501845769449844348729009139666901, 362360900703098064751535262345226972956088625684683815735347125467843, 230507868028808561984115998077450396305584227923739002411626810060636229, 146632478808056618968793667721035601663393840506352266080900143852973247507 , 93277006324123094477421357444997868881992870574852455865611857337978472\ 684341, 59336103267330617445347305823872628563389034645799776952862686675\ 277742184890083, 37745348938305512708505962942160765896503880046643070641\ 796636875233055100958664869 ----------------------------- This took, 1004.485, seconds. Theorem Number, 14 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 14 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- \ n ) B(n) t = - ( / ----- n = 0 6 5 4 3 2 / 8 t + 63044 t - 2116566 t - 22761161 t + 385463 t + 621 t - 1) / ( / 6 5 4 3 2 (t + 1) (8 t - 2500 t - 831478 t + 25678925 t - 415739 t - 625 t + 1)) and in Maple notation -(8*t^6+63044*t^5-2116566*t^4-22761161*t^3+385463*t^2+621*t-1)/(t+1)/(8*t^6-\ 2500*t^5-831478*t^4+25678925*t^3-415739*t^2-625*t+1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 32773, 19197419, 25526166389, 23093417825067, 24552667175438357, 24290782867918304971, 24796247671468698206581, 24965813676178062792344043, 25288659981903786348909940693, 25547954113686327708368259955979, 25839878893911250427748444801371189, 26121840359525377291658111082716916267, 26412772665617176235336349304356162390677, 26704331635738638010460139880791023098061131, 27000267673468135422884513316265375956477242101, 27298969537305674670167329382887311337598570615403, 27601203675571450872574654495343267841133045494152533, 27906682949265459481026335434553360799961168492739349899, 28215587917038879685750617762420009426404150796395576512949, 28527892370259342439075295313090260756955683842831575570571947, 28843662367862826251632472883522312530951764558918408666578630677, 29162923663942895640157496761696799734579981023086427452013888643531, 29485720491243647983300674135168600918663280838589669719003516741484661, 29812089505866994539539323644490035233569426933086562186796010617853352683, 301420713458709175457630759692077150028239084697838781457325777745393274\ 70293, 304757055135849042912933812802544643611266702741400713531785356108\ 19929240493579, 308130326515379586555566150691796124658154045876146892770\ 67728376105459899077097269, 311540935404585025607436749182192131645496420\ 53119682743758195459022922386989257784107, 314989295506425987654766301976\ 31157931122899214333655545420714915864895226306183651607957 ----------------------------- This took, 3664.174, seconds. Theorem Number, 15 --------------------------------- Let n - 1 --------' // i \2 i \ ' | | || (2 )| (2 ) | F[n](x) = | | \\x / + x + 1/ | | | | i = 0 Write: infinity ----- \ i F[n](x) = ) a(n, i) x / ----- i = 0 Let : infinity ----- \ 15 B(n) = ) a(n, k) / ----- k = 0 Then infinity ----- \ n ) B(n) t = / ----- n = 0 5 4 3 2 16637076 t + 134208623 t - 119408756 t + 994458 t + 1000 t - 1 ------------------------------------------------------------------- 5 4 3 2 163217943 t + 31379507 t - 130790642 t + 1056990 t + 1003 t - 1 and in Maple notation (16637076*t^5+134208623*t^4-119408756*t^3+994458*t^2+1000*t-1)/(163217943*t^5+ 31379507*t^4-130790642*t^3+1056990*t^2+1003*t-1) For the sake of the OEIS, here are the first 30 terms, starting at n=0 1, 3, 65541, 57526707, 126480267669, 179092953756003, 305796712933973541, 479472944711744914707, 780715727028372364191669, 1249866274126645791838008003, 2016123639981543018567911657541, 3241172948369208763467974713198707, 5218472757764444609436917831742771669, 8396364813124258088935079513778897876003, 13513575806009305458058879558434690690557541, 21746564810480795650299839311986402134518858707, 34997517333851295190900313637309593816113655367669, 56321226495036137862248316036031945017835289927280003, 90638394268155328363254754993440623428856311806208913541, 145864619485588244544019352123124157700524068214878849574707, 234740902084299059411038797810254011395117214677861978962139669, 377769688208986558547510000919771272763230010446650320157033740003, 607946909520853126875362736087098101024728776862880457427107091365541, 978372211655828246298777400622103094630092237156940902130720642375426707, 157449978199775238494422027228259851463768072131227587790706844323894244\ 7669, 2533851051284193965177543432951745435376305942722004233034682819661\ 665099176003, 40777403380675195562595971869148831188331064530432151817431\ 73514632292687722553541, 656232974452776450194231041199429473684673228697\ 2810041363631644116015474427728094707, 1056079305468033120983073037028725\ 1566451896809961509357907274672369735619389640133251669, 1699554184177104\ 248360257194945417104677593163940852081431441914677651688663409012996870\ 8003, 2735101816891187562158855402786045017959638449912852201562122649137\ 8566204498064865777232717541 ----------------------------- This took, 23185.590, seconds. ----------------------------------------- This concludes this article that took, 28223.139, seconds to produce.