The Number of Domino Tilings of Crosses with equal arm-lengths whose central rectangle has dimensions up to, 4 By Shalosh B. Ekhad [ShaloshBEkhad@gmail.com ] ------------------------------------------------------------------- Theorem number, 1, : Let A(n) be the number of ways to tile, with dominoes, the region in the plane consisting of a cross whose center is an, 1, by , 2, rectangle and that each of the four arms has length exacty n. Then infinity ----- 2 2 \ n (1 + t ) (t - t - 1) ) A(n) t = --------------------------------------------- / 2 2 ----- (t - 1) (t + 1) (t + 3 t + 1) (t - 3 t + 1) n = 0 and in Maple input format: (1+t^2)*(t^2-t-1)/(t-1)/(t+1)/(t^2+3*t+1)/(t^2-3*t+1) Equivalently, A(n) satisfies the linear recurrence A(n) = 8 A(n - 2) - 8 A(n - 4) + A(n - 6) subject to the initial conditions A(0) = 1, A(1) = 1, A(2) = 8, A(3) = 9, A(4) = 55, A(5) = 64 For the sake of Sloane here are the first 31 terms, starting at n=0 [1, 1, 8, 9, 55, 64, 377, 441, 2584, 3025, 17711, 20736, 121393, 142129, 832040, 974169, 5702887, 6677056, 39088169, 45765225, 267914296, 313679521, 1836311903, 2149991424, 12586269025, 14736260449, 86267571272, 101003831721, 591286729879, 692290561600] ------------------------------------------------------------------- Theorem number, 2, : Let A(n) be the number of ways to tile, with dominoes, the region in the plane consisting of a cross whose center is an, 1, by , 4, rectangle and that each of the four arms has length exacty n. Then infinity ----- \ n 2 3 4 5 6 7 ) A(n) t = - (-1 - t - t - 49 t + 503 t + 46 t - 2280 t + 694 t / ----- n = 0 8 9 10 11 12 13 14 / + 2280 t + 46 t - 503 t - 49 t + t - t + t ) / ( / 4 3 2 4 3 2 (t + 7 t + 13 t + 7 t + 1) (t - 7 t + 13 t - 7 t + 1) 4 3 2 4 3 2 (t - 11 t + 25 t - 11 t + 1) (t + 11 t + 25 t + 11 t + 1)) and in Maple input format: -(-1-t-t^2-49*t^3+503*t^4+46*t^5-2280*t^6+694*t^7+2280*t^8+46*t^9-503*t^10-49*t ^11+t^12-t^13+t^14)/(t^4+7*t^3+13*t^2+7*t+1)/(t^4-7*t^3+13*t^2-7*t+1)/(t^4-11*t ^3+25*t^2-11*t+1)/(t^4+11*t^3+25*t^2+11*t+1) Equivalently, A(n) satisfies the linear recurrence A(n) = 94 A(n - 2) - 2091 A(n - 4) + 14132 A(n - 6) - 31373 A(n - 8) + 14132 A(n - 10) - 2091 A(n - 12) + 94 A(n - 14) - A(n - 16) subject to the initial conditions A(0) = 1, A(1) = 1, A(2) = 95, A(3) = 143, A(4) = 6336, A(5) = 11305, A(6) = 413351, A(7) = 777095, A(8) = 26915305, A(9) = 51397632, A(10) = 1752296281, A(11) = 3361761865, A(12) = 114079985111, A(13) = 219162420359, A(14) = 7426955448000, A(15) = 14273954451361 For the sake of Sloane here are the first 31 terms, starting at n=0 [1, 1, 95, 143, 6336, 11305, 413351, 777095, 26915305, 51397632, 1752296281, 3361761865, 114079985111, 219162420359, 7426955448000, 14273954451361, 483517428660911, 929389976505935, 31478457514091281, 60508278530777088, 2049343473222853105, 3939314883731110225, 133418502766376903951, 256462201216559848895, 8685950946188033555136, 16696485994510535305177, 565481865522436582872695, 1086992396767612784594519, 36814592002163544002561401, 70766521944358488716160000] ------------------------------------------------------------------- Theorem number, 3, : Let A(n) be the number of ways to tile, with dominoes, the region in the plane consisting of a cross whose center is an, 2, by , 3, rectangle and that each of the four arms has length exacty n. Then infinity ----- \ n 2 3 4 5 6 ) A(n) t = (-3 - 13 t + 87 t - 9 t - 544 t + 1136 t + 12 t / ----- n = 0 7 8 9 10 11 12 13 14 - 2878 t + 1354 t + 2512 t - 304 t - 544 t - 9 t - 9 t - 5 t 15 / 2 2 2 + t ) / ((t - 1) (t + 1) (t - 4 t + 1) (t + 4 t + 1) (t + 3 t + 1) / 2 4 3 2 (t - 3 t + 1) (t + 12 t + 23 t + 12 t + 1) 4 3 2 (t - 12 t + 23 t - 12 t + 1)) and in Maple input format: (-3-13*t+87*t^2-9*t^3-544*t^4+1136*t^5+12*t^6-2878*t^7+1354*t^8+2512*t^9-304*t^ 10-544*t^11-9*t^12-9*t^13-5*t^14+t^15)/(t-1)/(t+1)/(t^2-4*t+1)/(t^2+4*t+1)/(t^2 +3*t+1)/(t^2-3*t+1)/(t^4+12*t^3+23*t^2+12*t+1)/(t^4-12*t^3+23*t^2-12*t+1) Equivalently, A(n) satisfies the linear recurrence A(n) = 120 A(n - 2) - 2520 A(n - 4) + 17423 A(n - 6) - 43440 A(n - 8) + 43440 A(n - 10) - 17423 A(n - 12) + 2520 A(n - 14) - 120 A(n - 16) + A(n - 18) subject to the initial conditions A(0) = 3, A(1) = 13, A(2) = 273, A(3) = 1569, A(4) = 25744, A(5) = 154384, A(6) = 2453577, A(7) = 14801577, A(8) = 234179165, A(9) = 1413911015, A(10) = 22355294976, A(11) = 134991588096, A(12) = 2134146060891, A(13) = 12887184179701, A(14) = 203736817744761, A(15) = 1230281569808697, A(16) = 19449799752900016, A(17) = 117449262683726896 For the sake of Sloane here are the first 31 terms, starting at n=0 [3, 13, 273, 1569, 25744, 154384, 2453577, 14801577, 234179165, 1413911015, 22355294976, 134991588096, 2134146060891, 12887184179701, 203736817744761, 1230281569808697, 19449799752900016, 117449262683726896, 1856781432495448305, 11212332504300826305, 177258242832917712821, 1070388974500414820495, 16922015782191958201344, 102185031572441386619904, 1615465738721627824782579, 9755127265441364921439709, 154220962009891983424585185, 931276396262380092689022609, 14722754282737731647837371600, 88904603971111966098059851600] ------------------------------------------------------------------- Theorem number, 4, : Let A(n) be the number of ways to tile, with dominoes, the region in the plane consisting of a cross whose center is an, 2, by , 4, rectangle and that each of the four arms has length exacty n. Then infinity ----- \ n 2 3 4 5 6 ) A(n) t = - (-5 + 525 t - 1028 t - 9215 t + 46001 t - 8332 t / ----- n = 0 7 8 9 10 11 - 414967 t + 522954 t + 1439705 t - 1846474 t - 2457886 t 12 13 14 15 16 + 1910156 t + 1631814 t - 576334 t - 274621 t - 1354 t 17 18 19 20 21 22 23 - 5641 t + 29688 t + 5399 t - 2513 t - 448 t - 61 t + 8 t 24 / 2 4 3 2 + t ) / ((t + 1) (t - 3 t + 1) (t + 13 t + 18 t + 8 t + 1) / 4 3 2 4 3 2 (t + 11 t + 25 t + 11 t + 1) (t + 8 t + 18 t + 13 t + 1) 4 3 2 (t - 7 t + 13 t - 7 t + 1) 8 7 6 5 4 3 2 (t - 33 t + 296 t - 1023 t + 1519 t - 1023 t + 296 t - 33 t + 1)) and in Maple input format: -(-5+525*t^2-1028*t^3-9215*t^4+46001*t^5-8332*t^6-414967*t^7+522954*t^8+1439705 *t^9-1846474*t^10-2457886*t^11+1910156*t^12+1631814*t^13-576334*t^14-274621*t^ 15-1354*t^16-5641*t^17+29688*t^18+5399*t^19-2513*t^20-448*t^21-61*t^22+8*t^23+t ^24)/(t+1)/(t^2-3*t+1)/(t^4+13*t^3+18*t^2+8*t+1)/(t^4+11*t^3+25*t^2+11*t+1)/(t^ 4+8*t^3+18*t^2+13*t+1)/(t^4-7*t^3+13*t^2-7*t+1)/(t^8-33*t^7+296*t^6-1023*t^5+ 1519*t^4-1023*t^3+296*t^2-33*t+1) Equivalently, A(n) satisfies the linear recurrence A(n) = 10 A(n - 1) + 330 A(n - 2) - 1089 A(n - 3) - 23200 A(n - 4) + 57320 A(n - 5) + 557579 A(n - 6) - 1264230 A(n - 7) - 5255350 A(n - 8) + 10463111 A(n - 9) + 23656280 A(n - 10) - 37716240 A(n - 11) - 54766801 A(n - 12) + 64374650 A(n - 13) + 64374650 A(n - 14) - 54766801 A(n - 15) - 37716240 A(n - 16) + 23656280 A(n - 17) + 10463111 A(n - 18) - 5255350 A(n - 19) - 1264230 A(n - 20) + 557579 A(n - 21) + 57320 A(n - 22) - 23200 A(n - 23) - 1089 A(n - 24) + 330 A(n - 25) + 10 A(n - 26) - A(n - 27) subject to the initial conditions A(0) = 5, A(1) = 50, A(2) = 1625, A(3) = 28333, A(4) = 658345, A(5) = 13244314, A(6) = 286804580, A(7) = 5979523882, A(8) = 127184190505, A(9) = 2676754469497, A(10) = 56655287134441, A(11) = 1195514498552642, A(12) = 25268522461460609, A(13) = 533606256925040200, A(14) = 11273790075763916200, A(15) = 238125664699753375565, A(16) = 5030413511261988529170, A(17) = 106259535498363551100985, A(18) = 2244658170103868422356057, A(19) = 47415756907202301755608185, A(20) = 1001614274851131970960044866, A(21) = 21158039172191811496855827860, A(22) = 446942762720405765476672607738, A(23) = 9441207474296162613101221394905, A(24) = 199436039775327403021485786534013, A(25) = 4212883866394267951131950209331569, A(26) = 88992923003077943934208907901818498 For the sake of Sloane here are the first 31 terms, starting at n=0 [5, 50, 1625, 28333, 658345, 13244314, 286804580, 5979523882, 127184190505, 2676754469497, 56655287134441, 1195514498552642, 25268522461460609, 533606256925040200, 11273790075763916200, 238125664699753375565, 5030413511261988529170, 106259535498363551100985, 2244658170103868422356057, 47415756907202301755608185, 1001614274851131970960044866, 21158039172191811496855827860, 446942762720405765476672607738, 9441207474296162613101221394905, 199436039775327403021485786534013, 4212883866394267951131950209331569, 88992923003077943934208907901818498, 1879885423293773731793287610693140081, 39710680569746944489720740240637955200, 838847969546591713371662943119398787200] ------------------------------------------------------------------- Theorem number, 5, : Let A(n) be the number of ways to tile, with dominoes, the region in the plane consisting of a cross whose center is an, 2, by , 5, rectangle and that each of the four arms has length exacty n. Then infinity ----- \ n 2 3 4 5 ) A(n) t = - 8 (-1 - 20 t + 275 t + 520 t - 15210 t + 32028 t / ----- n = 0 6 7 8 9 10 + 233881 t - 443920 t - 1247585 t + 728540 t + 2438268 t 11 12 13 14 15 + 1005864 t - 996215 t - 1285420 t + 264745 t + 311768 t 16 17 18 19 20 21 / - 96594 t + 11580 t - 865 t - 1520 t - 91 t + 4 t ) / ( / 2 2 2 2 (t + 1) (t + 5 t + 1) (t - 5 t + 1) (t + 7 t + 1) (t - 3 t + 1) 4 3 2 4 3 2 (t - 35 t + 72 t - 35 t + 1) (t + 35 t + 72 t + 35 t + 1) 4 3 2 4 3 2 (t + 15 t + 32 t + 15 t + 1) (t - 15 t + 32 t - 15 t + 1)) and in Maple input format: -8*(-1-20*t+275*t^2+520*t^3-15210*t^4+32028*t^5+233881*t^6-443920*t^7-1247585*t ^8+728540*t^9+2438268*t^10+1005864*t^11-996215*t^12-1285420*t^13+264745*t^14+ 311768*t^15-96594*t^16+11580*t^17-865*t^18-1520*t^19-91*t^20+4*t^21)/(t+1)/(t^2 +5*t+1)/(t^2-5*t+1)/(t^2+7*t+1)/(t^2-3*t+1)/(t^4-35*t^3+72*t^2-35*t+1)/(t^4+35* t^3+72*t^2+35*t+1)/(t^4+15*t^3+32*t^2+15*t+1)/(t^4-15*t^3+32*t^2-15*t+1) Equivalently, A(n) satisfies the linear recurrence A(n) = -5 A(n - 1) + 1280 A(n - 2) + 6340 A(n - 3) - 224900 A(n - 4) - 1048576 A(n - 5) + 8239880 A(n - 6) + 28813840 A(n - 7) - 104783360 A(n - 8) - 210289420 A(n - 9) + 470841548 A(n - 10) + 635787520 A(n - 11) - 855213330 A(n - 12) - 855213330 A(n - 13) + 635787520 A(n - 14) + 470841548 A(n - 15) - 210289420 A(n - 16) - 104783360 A(n - 17) + 28813840 A(n - 18) + 8239880 A(n - 19) - 1048576 A(n - 20) - 224900 A(n - 21) + 6340 A(n - 22) + 1280 A(n - 23) - 5 A(n - 24) - A(n - 25) subject to the initial conditions A(0) = 8, A(1) = 120, A(2) = 7440, A(3) = 162960, A(4) = 7791680, A(5) = 181167168, A(6) = 8365643832, A(7) = 196236745320, A(8) = 9016149528320, A(9) = 211768687199520, A(10) = 9722665019273472, A(11) = 228406050634035456, A(12) = 10485398166402971240, A(13) = 246331027535352254040, A(14) = 11308101362898799991280, A(15) = 265659665840429104758192, A(16) = 12195376479742968698940992, A(17) = 286504465038364584457596480, A(18) = 13152273784719106978244904600, A(19) = 308984760897111556636355862600, A(20) = 14184253547282862782655960793568, A(21) = 333228939497877815806797193366080, A(22) = 15297206632882501156906516391347200, A(23) = 359375411318181303865384238399308800, A(24) = 16497486467857782619428192588905395400 For the sake of Sloane here are the first 31 terms, starting at n=0 [8, 120, 7440, 162960, 7791680, 181167168, 8365643832, 196236745320, 9016149528320, 211768687199520, 9722665019273472, 228406050634035456, 10485398166402971240, 246331027535352254040, 11308101362898799991280, 265659665840429104758192, 12195376479742968698940992, 286504465038364584457596480, 13152273784719106978244904600, 308984760897111556636355862600, 14184253547282862782655960793568, 333228939497877815806797193366080, 15297206632882501156906516391347200, 359375411318181303865384238399308800, 16497486467857782619428192588905395400, 387573439318406416843926037663295706168, 17791945047228835162623284479070980197072, 417983997005822122047833426727136098340560, 19187972009290560675874087886737815925520960, 450780688319034887389260798770063492988893760] ------------------------------------------------------------------- Theorem number, 6, : Let A(n) be the number of ways to tile, with dominoes, the region in the plane consisting of a cross whose center is an, 3, by , 4, rectangle and that each of the four arms has length exacty n. Then infinity ----- \ n 2 3 4 5 ) A(n) t = - (11 - 5258 t + 14283 t + 683050 t - 3802521 t / ----- n = 0 6 7 8 9 - 30981728 t + 279433657 t - 150068000 t - 8978851647 t 10 11 12 + 53351993825 t + 136953010980 t - 1612105188442 t 13 14 16 - 807376752098 t + 23209741235614 t - 197804407767450 t 17 18 19 + 72487871451366 t + 1079855221286174 t - 467260852723820 t 20 21 22 - 3883115687999216 t + 1640990270982300 t + 9230493573538150 t 23 24 15 - 3594086136926130 t - 14373520285994430 t - 2990222356916 t 25 26 27 + 5177769134790000 t + 14418503987874130 t - 4963751516205030 t 28 29 30 - 9080370629599850 t + 3109027484844420 t + 3413244740295184 t 31 32 33 - 1231458575743276 t - 656538976426242 t + 297053936770362 t 34 35 36 + 9714953833350 t - 41150066600724 t + 23475569046398 t 37 38 39 + 2900618684018 t - 5013205763130 t - 79161363228 t 40 41 42 43 + 465390458825 t + 1040888895 t - 20312260088 t - 43397857 t 44 45 46 47 48 49 + 386605176 t - 2671999 t - 2771150 t + 95589 t + 46 t - 685 t 50 / 2 2 + 51 t + 61 t) / ((t - 1) (t + 1) (t - 4 t + 1) (t + 4 t + 1) / 4 3 2 4 3 2 (t - 11 t + 25 t - 11 t + 1) (t + 11 t + 25 t + 11 t + 1) 4 3 2 4 3 2 (t + 7 t + 13 t + 7 t + 1) (t - 7 t + 13 t - 7 t + 1) 8 7 6 5 4 3 2 (t + 28 t + 231 t + 728 t + 1049 t + 728 t + 231 t + 28 t + 1) 8 7 6 5 4 3 2 (t - 28 t + 231 t - 728 t + 1049 t - 728 t + 231 t - 28 t + 1) 8 7 6 5 4 3 2 (t + 44 t + 471 t + 1672 t + 2513 t + 1672 t + 471 t + 44 t + 1) 8 7 6 5 4 3 2 (t - 44 t + 471 t - 1672 t + 2513 t - 1672 t + 471 t - 44 t + 1)) and in Maple input format: -(11-5258*t^2+14283*t^3+683050*t^4-3802521*t^5-30981728*t^6+279433657*t^7-\ 150068000*t^8-8978851647*t^9+53351993825*t^10+136953010980*t^11-1612105188442*t ^12-807376752098*t^13+23209741235614*t^14-197804407767450*t^16+72487871451366*t ^17+1079855221286174*t^18-467260852723820*t^19-3883115687999216*t^20+ 1640990270982300*t^21+9230493573538150*t^22-3594086136926130*t^23-\ 14373520285994430*t^24-2990222356916*t^15+5177769134790000*t^25+ 14418503987874130*t^26-4963751516205030*t^27-9080370629599850*t^28+ 3109027484844420*t^29+3413244740295184*t^30-1231458575743276*t^31-\ 656538976426242*t^32+297053936770362*t^33+9714953833350*t^34-41150066600724*t^ 35+23475569046398*t^36+2900618684018*t^37-5013205763130*t^38-79161363228*t^39+ 465390458825*t^40+1040888895*t^41-20312260088*t^42-43397857*t^43+386605176*t^44 -2671999*t^45-2771150*t^46+95589*t^47+46*t^48-685*t^49+51*t^50+61*t)/(t-1)/(t+1 )/(t^2-4*t+1)/(t^2+4*t+1)/(t^4-11*t^3+25*t^2-11*t+1)/(t^4+11*t^3+25*t^2+11*t+1) /(t^4+7*t^3+13*t^2+7*t+1)/(t^4-7*t^3+13*t^2-7*t+1)/(t^8+28*t^7+231*t^6+728*t^5+ 1049*t^4+728*t^3+231*t^2+28*t+1)/(t^8-28*t^7+231*t^6-728*t^5+1049*t^4-728*t^3+ 231*t^2-28*t+1)/(t^8+44*t^7+471*t^6+1672*t^5+2513*t^4+1672*t^3+471*t^2+44*t+1)/ (t^8-44*t^7+471*t^6-1672*t^5+2513*t^4-1672*t^3+471*t^2-44*t+1) Equivalently, A(n) satisfies the linear recurrence A(n) = 1425 A(n - 2) - 561450 A(n - 4) + 90789794 A(n - 6) - 7424197275 A(n - 8) + 336789859425 A(n - 10) - 8907131413306 A(n - 12) + 143455495753800 A(n - 14) - 1462873333184025 A(n - 16) + 9771717521515319 A(n - 18) - 43829101525069575 A(n - 20) + 134037886600289925 A(n - 22) - 281699897521436101 A(n - 24) + 408167637282649875 A(n - 26) - 408167637282649875 A(n - 28) + 281699897521436101 A(n - 30) - 134037886600289925 A(n - 32) + 43829101525069575 A(n - 34) - 9771717521515319 A(n - 36) + 1462873333184025 A(n - 38) - 143455495753800 A(n - 40) + 8907131413306 A(n - 42) - 336789859425 A(n - 44) + 7424197275 A(n - 46) - 90789794 A(n - 48) + 561450 A(n - 50) - 1425 A(n - 52) + A(n - 54) subject to the initial conditions A(0) = 11, A(1) = 61, A(2) = 10417, A(3) = 101208, A(4) = 9351325, A(5) = 106170429, A(6) = 8444719481, A(7) = 100287240816, A(8) = 7647364885248, A(9) = 92026729386480, A(10) = 6931632256681825, A(11) = 83740302374020131, A(12) = 6284639960849143981, A(13) = 76011893079484863240, A(14) = 5698512679975989078249, A(15) = 68946354812578119870331, A(16) = 5167177662565655884138075, A(17) = 62524062928398036242403456, A(18) = 4685419250169132757337358336, A(19) = 56696373833881835246682179328, A(20) = 4248586503525479251030717807875, A(21) = 51410893066069945853289050331461, A(22) = 3852483189595276525717941461080825, A(23) = 46617886188880913996887505531954424, A(24) = 3493309961419648587573268882218641221, A(25) = 42271658295103555449311167405258765685, A(26) = 3167623211979350348305670779900790277681, A(27) = 38330614335318787827327920867599519446480, A(28) = 2872300795378119546949465765399491372000000, A(29) = 34756994183721320297083158848755497524331600, A(30) = 2604511757816504926966701583386418305386457161, A(31) = 31516546592975249135336325251919206718824091451, A(32) = 2361689109111573680647210494143074676982814157797, A(33) = 28578210509979859801708368698601055397560359786408, A(34) = 2141505191433676308012143641546335718875683396677825, A(35) = 25913819918657122747443135423977789635087143738482019, A(36) = 1941849360202688446474368570301343667723341041995950531, A(37) = 23497834540453670235109586087985956414837825663709509376, A(38) = 1760807750056603521797378512632679821707301397079283990528, A(39) = 21307095196344831804918566868076287440307981368720172979200, A(40) = 1596644928405082149007362025637722192564806506499204784150075, A(41) = 19320601857163455398087271097000350267014448224609793522635661, A(42) = 1447787259755194896559288096462120214533130108532797043814159681, A(43) = 17519312354517161883750385628711963790458323531409341286817184920, A(44) = 1312807821088526910821906975259844639187323145374970622659243726829 , A(45) = 15885959849573878702825368281849091779399456274147845466825970749421, A(46) = 1190412723622894638560563481275186496834406406768542633110883597574825, A(47) = 14404887317198972423319613936253579581780385125004926299524823923228976, A(48) = 1079428709823204288359840023817078628918766393649351774033387221192839936, A(49) = 13061897460770022653742874427759076611022801960442839906696590947950526128, A(50) = 978791906763695559141680633472942913173220504508288829397309049156108497905 , A(51) = 11844116619498001840207787008222152739485708432623206132590250\ 159749846871571, A(52) = 88753762803199018519084775476061616300238296700\ 7969561074495181557775073399325, A(53) = 1073987136383396266211088535756\ 1549686115685724083372754828238785171699754368584 For the sake of Sloane here are the first 31 terms, starting at n=0 [11, 61, 10417, 101208, 9351325, 106170429, 8444719481, 100287240816, 7647364885248, 92026729386480, 6931632256681825, 83740302374020131, 6284639960849143981, 76011893079484863240, 5698512679975989078249, 68946354812578119870331, 5167177662565655884138075, 62524062928398036242403456, 4685419250169132757337358336, 56696373833881835246682179328, 4248586503525479251030717807875, 51410893066069945853289050331461, 3852483189595276525717941461080825, 46617886188880913996887505531954424, 3493309961419648587573268882218641221, 42271658295103555449311167405258765685, 3167623211979350348305670779900790277681, 38330614335318787827327920867599519446480, 2872300795378119546949465765399491372000000, 34756994183721320297083158848755497524331600, 2604511757816504926966701583386418305386457161, 31516546592975249135336325251919206718824091451, 2361689109111573680647210494143074676982814157797, 28578210509979859801708368698601055397560359786408, 2141505191433676308012143641546335718875683396677825, 25913819918657122747443135423977789635087143738482019, 1941849360202688446474368570301343667723341041995950531, 23497834540453670235109586087985956414837825663709509376, 1760807750056603521797378512632679821707301397079283990528, 21307095196344831804918566868076287440307981368720172979200, 1596644928405082149007362025637722192564806506499204784150075, 19320601857163455398087271097000350267014448224609793522635661, 1447787259755194896559288096462120214533130108532797043814159681, 17519312354517161883750385628711963790458323531409341286817184920, 1312807821088526910821906975259844639187323145374970622659243726829, 15885959849573878702825368281849091779399456274147845466825970749421, 1190412723622894638560563481275186496834406406768542633110883597574825, 14404887317198972423319613936253579581780385125004926299524823923228976, 1079428709823204288359840023817078628918766393649351774033387221192839936, 13061897460770022653742874427759076611022801960442839906696590947950526128, 978791906763695559141680633472942913173220504508288829397309049156108497905 , 11844116619498001840207787008222152739485708432623206132590250159749846\ 871571, 88753762803199018519084775476061616300238296700796956107449518155\ 7775073399325, 1073987136383396266211088535756154968611568572408337275482\ 8238785171699754368584] ------------------------------------------------------------------- I hope, dear reader, that you enjoined reading these, 6, theorems as much as I did discovering them. This ends this fascinating book! It took, 608.334, seconds to generate it.