{[x,0],[0,1], x is a power of 4} Number of walks sequence [0, 0, 0, 5, 0, 0, 0, 45, 0, 0, 0, 455, 0, 0, 0, 4862, 0, 0, 0, 53592, 0, 0, 0, 602550, 0, 0, 0, 6868360, 0, 0, 0, 79084731, 0, 0, 0, 917665268, 0, 0, 0, 10713333169, 0, 0, 0, 125692939890, 0, 0, 0, 1480714435550, 0, 0] Number of Good walks sequence [0, 0, 0, 1, 0, 0, 0, 5, 0, 0, 0, 35, 0, 0, 0, 286, 0, 0, 0, 2552, 0, 0, 0, 24102, 0, 0, 0, 236840, 0, 0, 0, 2396507, 0, 0, 0, 24801764, 0, 0, 0, 261300809, 0, 0, 0, 2793176442, 0, 0, 0, 30218661950, 0, 0] {[1,2],[2,1]} Chess Knight Number of walks sequence [0, 0, 2, 0, 0, 6, 0, 0, 20, 0, 0, 70, 0, 0, 252, 0, 0, 924, 0, 0, 3432, 0, 0, 12870, 0, 0, 48620, 0, 0, 184756, 0, 0, 705432, 0, 0, 2704156, 0, 0, 10400600, 0, 0, 40116600, 0, 0, 155117520, 0, 0, 601080390, 0, 0] Number of Good walks sequence [0, 0, 1, 0, 0, 2, 0, 0, 5, 0, 0, 14, 0, 0, 42, 0, 0, 132, 0, 0, 429, 0, 0, 1430, 0, 0, 4862, 0, 0, 16796, 0, 0, 58786, 0, 0, 208012, 0, 0, 742900, 0, 0, 2674440, 0, 0, 9694845, 0, 0, 35357670, 0, 0] {[x,0],[0,1], x is a power of 3} Number of walks sequence [0, 0, 4, 0, 0, 28, 0, 0, 230, 0, 0, 2002, 0, 0, 17952, 0, 0, 164046, 0, 0, 1519012, 0, 0, 14203800, 0, 0, 133818440, 0, 0, 1268291840, 0, 0, 12078921022, 0, 0, 115499926310, 0, 0, 1108164818200, 0, 0, 10663040583312, 0, 0, 102858641485968, 0, 0, 994363671194670, 0, 0] Number of Good walks sequence [0, 0, 1, 0, 0, 4, 0, 0, 23, 0, 0, 154, 0, 0, 1122, 0, 0, 8634, 0, 0, 69046, 0, 0, 568152, 0, 0, 4779230, 0, 0, 40912640, 0, 0, 355262383, 0, 0, 3121619630, 0, 0, 27704120455, 0, 0, 247977687984, 0, 0, 2236057423608, 0, 0, 20293136146830, 0, 0] {[1, 2], [2, 1], [2, 2]} Number of walks sequence [0, 1, 2, 1, 6, 7, 12, 31, 40, 91, 170, 281, 602, 1051, 1988, 3907, 7044, 13735, 25962, 48643, 94094, 177145, 338184, 647791, 1228812, 2356927, 4500678, 8595913, 16486966, 31521543, 60419872, 115870879, 222045160, 426275647, 818054654, 1570608819, 3017824226, 5797766021, 11145588252, 21432188633, 41219486504, 79312241045, 152632815374, 293818488931, 565768439446, 1089621011617, 2099089036652, 4044617798711, 7794886185140, 15025880176531] Number of Good walks sequence [0, 1, 1, 1, 3, 3, 6, 11, 15, 31, 50, 85, 161, 267, 490, 883, 1548, 2863, 5127, 9307, 17116, 31021, 57123, 104963, 192699, 356643, 658034, 1218517, 2262079, 4196895, 7812028, 14549655, 27126118, 50671255, 94697293, 177220411, 332015747, 622464929, 1168361604, 2194661957, 4125875271, 7763334489, 14617493348, 27544491295, 51939746688, 98003747593, 185045277941, 349600991071, 660883094870, 1250051073411] {[x,0],[0,1], x is a power of 2} Number of walks sequence [0, 3, 0, 20, 0, 140, 0, 1044, 0, 7997, 0, 62426, 0, 493680, 0, 3942062, 0, 31712596, 0, 256628988, 0, 2086728820, 0, 17035559450, 0, 139541908086, 0, 1146299600818, 0, 9439903942400, 0, 77906914698294, 0, 644185678384220, 0, 5335552771987680, 0, 44259112207553568, 0, 367632973794879772, 0, 3057439536274024253, 0, 25455638149113884940, 0, 212153544035373537520, 0, 1769781534927760858560, 0, 14776088263031125765020] Number of Good walks sequence [0, 1, 0, 4, 0, 20, 0, 116, 0, 727, 0, 4802, 0, 32912, 0, 231886, 0, 1669084, 0, 12220428, 0, 90727340, 0, 681422378, 0, 5168218818, 0, 39527572442, 0, 304513030400, 0, 2360815596918, 0, 18405305096692, 0, 144204128972640, 0, 1134849030962912, 0, 8966657897436092, 0, 71103245029628471, 0, 565680847758086332, 0, 4513905192241990160, 0, 36117990508729813440, 0, 289727220843747564020] {[x,0],[0,1], x is a prime number} Number of walks sequence [0, 3, 4, 15, 48, 112, 440, 1080, 3850, 10923, 34956, 108381, 332976, 1060140, 3264272, 10374726, 32374080, 102332461, 322497780, 1017881403, 3222747902, 10188669260, 32317822704, 102429843750, 325252097196, 1033093503018, 3284112853084, 10447798281221, 33252542802270, 105913397249848, 337482546297632, 1075995331596207, 3432060653062868, 10952185181636300, 34964751031527024, 111667862139572535, 356775380921860156, 1140280355326239582, 3645680403770887120, 11659585914475970557, 37301061735064131186, 119367339890632572321, 382093874592706700864, 1223404369274091439485, 3918136042833876793378, 12551463905149736008988, 40217004031096503040128, 128890697053895752711050, 413167264096864937213950, 1324704533684286734855625] Number of Good walks sequence [0, 1, 1, 3, 8, 16, 55, 120, 385, 993, 2913, 8337, 23784, 70676, 204017, 610278, 1798560, 5385919, 16124889, 48470543, 146488541, 442985620, 1346575946, 4097193750, 12509696046, 38262722334, 117289744753, 360268906249, 1108418093409, 3416561201608, 10546329571801, 32605919139279, 100942960384202, 312919576618180, 971243084209084, 3018050328096555, 9388825813733162, 29237957828877938, 91142010094272178, 284380144255511477, 888120517501526933, 2775984648619362147, 8683951695288788656, 27186763761646476433, 85176870496388625943, 267052423513824170404, 837854250647843813336, 2630422388855015361450, 8263345281937298744279, 25974598699691896761875] {[1,1]} 1D reduction Number of walks sequence [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] Number of Good walks sequence [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] {[1,1],[2,2]} Fibonacci Number of walks sequence [1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141, 267914296, 433494437, 701408733, 1134903170, 1836311903, 2971215073, 4807526976, 7778742049, 12586269025, 20365011074] Number of Good walks sequence [1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141, 267914296, 433494437, 701408733, 1134903170, 1836311903, 2971215073, 4807526976, 7778742049, 12586269025, 20365011074] {[1,1],...,[i,i], i = 1..n} Chess Bishop Number of walks sequence [1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184, 34359738368, 68719476736, 137438953472, 274877906944, 549755813888, 1099511627776, 2199023255552, 4398046511104, 8796093022208, 17592186044416, 35184372088832, 70368744177664, 140737488355328, 281474976710656, 562949953421312] Number of Good walks sequence [1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184, 34359738368, 68719476736, 137438953472, 274877906944, 549755813888, 1099511627776, 2199023255552, 4398046511104, 8796093022208, 17592186044416, 35184372088832, 70368744177664, 140737488355328, 281474976710656, 562949953421312] {[1, 1], [1, 2], [2, 1], [2, 2]} Number of walks sequence [1, 2, 5, 11, 26, 63, 153, 376, 931, 2317, 5794, 14545, 36631, 92512, 234205, 594169, 1510192, 3844787, 9802895, 25027296, 63972861, 163701327, 419316330, 1075049011, 2758543201, 7083830648, 18204064403, 46812088751, 120452857976, 310118453573, 798864734705, 2058919125662, 5308995110801, 13695571200353, 35345294050448, 91254952276859, 235692238604699, 608960974528058, 1573913990936843, 4069232436916151, 10523909919751592, 27225039227088245, 70449962082969713, 182351090200262876, 472113236570698271, 1222614097529082473, 3166894060998873938, 8204920400637973901, 21262187978291212931, 55110019134351712292] Number of Good walks sequence [1, 2, 4, 8, 17, 37, 82, 185, 423, 978, 2283, 5373, 12735, 30372, 72832, 175502, 424748, 1032004, 2516347, 6155441, 15101701, 37150472, 91618049, 226460893, 560954047, 1392251012, 3461824644, 8622571758, 21511212261, 53745962199, 134474581374, 336908488839, 845139060165, 2122553644686, 5336735929371, 13432403613621, 33843022209066, 85349327734485, 215440028338359, 544288586926914, 1376230297675914, 3482537223611046, 8819175375714063, 22349794473772659, 56678600914995057, 143830921235537742, 365225623668676437, 927972354829010775, 2359192024476568203, 6001174121892988758] {[1, 1], [1, 2], [2, 0], [2, 1], [2, 2]} Number of walks sequence [1, 2, 5, 14, 38, 105, 293, 826, 2344, 6689, 19174, 55171, 159257, 460978, 1337509, 3888834, 11327704, 33050309, 96570740, 282545529, 827655599, 2427068998, 7124343910, 20931582711, 61549162531, 181124500448, 533386979573, 1571793965942, 4634650127822, 13673778498441, 40364002386085, 119211832783474, 352249721701344, 1041300246801325, 3079528374251948, 9110982366758033, 26965570686481751, 79837850052980810, 236458491437949252, 700552216604383117, 2076150413980940101, 6154651963871554746, 18250244382309227359, 54131353763954168144, 160597579823007945896, 476578258637923161153, 1414588429817781068294, 4199740209721592763871, 12471171770983493163931, 37040846455014934829892] Number of Good walks sequence [1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, 113634, 310572, 853467, 2356779, 6536382, 18199284, 50852019, 142547559, 400763223, 1129760415, 3192727797, 9043402501, 25669818476, 73007772802, 208023278209, 593742784829, 1697385471211, 4859761676391, 13933569346707, 40002464776083, 114988706524270, 330931069469828, 953467954114363, 2750016719520991, 7939655757745265, 22944749046030949, 66368199913921497, 192137918101841817, 556704809728838604, 1614282136160911722, 4684478925507420069, 13603677110519480289, 39532221379621112004, 114956499435014161638, 334496473194459009429, 973899740488107474693, 2837208756709314025578] {[0, 1], [1, 1], [2, 1]} Number of walks sequence [1, 3, 7, 19, 51, 141, 393, 1107, 3139, 8953, 25653, 73789, 212941, 616227, 1787607, 5196627, 15134931, 44152809, 128996853, 377379369, 1105350729, 3241135527, 9513228123, 27948336381, 82176836301, 241813226151, 712070156203, 2098240353907, 6186675630819, 18252025766941, 53876592856681, 159114492071763, 470139239360787, 1389754816243449, 4109922421017093, 12159131877715993, 35986168879543609, 106542797484006471, 315544068167601787, 934837217271732457, 2770417140954208377, 8212609533895771131, 24352194654450483759, 72228808291130603703, 214285636273290835239, 635888739568958641449, 1887427033736750828037, 5603455843421135356413, 16639279789182494873661, 49419934162239477797703] Number of Good walks sequence [1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, 113634, 310572, 853467, 2356779, 6536382, 18199284, 50852019, 142547559, 400763223, 1129760415, 3192727797, 9043402501, 25669818476, 73007772802, 208023278209, 593742784829, 1697385471211, 4859761676391, 13933569346707, 40002464776083, 114988706524270, 330931069469828, 953467954114363, 2750016719520991, 7939655757745265, 22944749046030949, 66368199913921497, 192137918101841817, 556704809728838604, 1614282136160911722, 4684478925507420069, 13603677110519480289, 39532221379621112004, 114956499435014161638, 334496473194459009429, 973899740488107474693, 2837208756709314025578] {[0, 2], [1, 1], [2, 0]} Number of walks sequence [1, 3, 7, 19, 51, 141, 393, 1107, 3139, 8953, 25653, 73789, 212941, 616227, 1787607, 5196627, 15134931, 44152809, 128996853, 377379369, 1105350729, 3241135527, 9513228123, 27948336381, 82176836301, 241813226151, 712070156203, 2098240353907, 6186675630819, 18252025766941, 53876592856681, 159114492071763, 470139239360787, 1389754816243449, 4109922421017093, 12159131877715993, 35986168879543609, 106542797484006471, 315544068167601787, 934837217271732457, 2770417140954208377, 8212609533895771131, 24352194654450483759, 72228808291130603703, 214285636273290835239, 635888739568958641449, 1887427033736750828037, 5603455843421135356413, 16639279789182494873661, 49419934162239477797703] Number of Good walks sequence [1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, 113634, 310572, 853467, 2356779, 6536382, 18199284, 50852019, 142547559, 400763223, 1129760415, 3192727797, 9043402501, 25669818476, 73007772802, 208023278209, 593742784829, 1697385471211, 4859761676391, 13933569346707, 40002464776083, 114988706524270, 330931069469828, 953467954114363, 2750016719520991, 7939655757745265, 22944749046030949, 66368199913921497, 192137918101841817, 556704809728838604, 1614282136160911722, 4684478925507420069, 13603677110519480289, 39532221379621112004, 114956499435014161638, 334496473194459009429, 973899740488107474693, 2837208756709314025578] {[0,1],[1,0]} Standard Number of walks sequence [2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, 2704156, 10400600, 40116600, 155117520, 601080390, 2333606220, 9075135300, 35345263800, 137846528820, 538257874440, 2104098963720, 8233430727600, 32247603683100, 126410606437752, 495918532948104, 1946939425648112, 7648690600760440, 30067266499541040, 118264581564861424, 465428353255261088, 1832624140942590534, 7219428434016265740, 28453041475240576740, 112186277816662845432, 442512540276836779204, 1746130564335626209832, 6892620648693261354600, 27217014869199032015600, 107507208733336176461620, 424784580848791721628840, 1678910486211891090247320, 6637553085023755473070800, 26248505381684851188961800, 103827421287553411369671120, 410795449442059149332177040, 1625701140345170250548615520, 6435067013866298908421603100, 25477612258980856902730428600, 100891344545564193334812497256] Number of Good walks sequence [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324, 4861946401452, 18367353072152, 69533550916004, 263747951750360, 1002242216651368, 3814986502092304, 14544636039226909, 55534064877048198, 212336130412243110, 812944042149730764, 3116285494907301262, 11959798385860453492, 45950804324621742364, 176733862787006701400, 680425371729975800390, 2622127042276492108820, 10113918591637898134020, 39044429911904443959240, 150853479205085351660700, 583300119592996693088040, 2257117854077248073253720, 8740328711533173390046320, 33868773757191046886429490, 131327898242169365477991900, 509552245179617138054608572, 1978261657756160653623774456] {[x^4,0],[0,1]} x to the 4th power Number of walks sequence [2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, 2704156, 10400600, 40116600, 155117520, 601080407, 2333606562, 9075139290, 35345299220, 137846794470, 538259650500, 2104109889180, 8233493837400, 32247950787000, 126412442231712, 495927937861776, 1946986355217344, 7648919661753120, 30068363804986080, 118269754576245184, 465452404977006656, 1832734628539359798, 7219930650365235810, 28455303090041187080, 112196378562468669612, 442557320249958781504, 1746327780639846933620, 6893484059789685332520, 27220774605841452846440, 107523500925653957277070, 424854867752001874161744, 1679212488188993143276044, 6638845892848760469728080, 26254020802375247204495340, 103850877676057569024213120, 410894918136339004946355400, 1626121817191674770728807200, 6436841744568966494898587075, 25485082153413730115022859000, 100922718108868798803841224486] Number of Good walks sequence [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357671, 129644809, 477638910, 1767264961, 6564133070, 24466347750, 91483038660, 343062243225, 1289918031480, 4862017008912, 18367701402288, 69535226972048, 263755850405280, 1002278793499536, 3815153373427264, 14545387655531458, 55537412986041206, 212350901481330465, 813008659715462488, 3116566071179685267, 11961008655404291392, 45955994227364392990, 176756001533068854680, 680519365146036321161, 2622524412820828226270, 10115592089333377956232, 39051453213697514959908, 150882861201108192493820, 583422684497227715655452, 2257627775566468891830720, 8742445066730617126518200, 33877537858159891056850150, 131364117236101357038746675, 509701643068274602300457180, 1978876825664094094192965186] {[x^3,0],[0,1]} x cubed Number of walks sequence [2, 6, 20, 70, 252, 924, 3432, 12879, 48730, 185614, 710892, 2735096, 10563392, 40930560, 159040032, 619467318, 2417972490, 9455749380, 37039135140, 145301145990, 570762170880, 2244725842020, 8837820660240, 34830440619975, 137394113691744, 542426109370362, 2143127632766260, 8473549410010694, 33525109130598720, 132721876368354976, 525731878184759840, 2083624087047093360, 8262148572917573628, 32777201466163682010, 130089976611473428956, 516532679439533660748, 2051746089353447680180, 8152896725414100558444, 32408146839747607260920, 128867221552876914210586, 512589163095719982535410, 2039521468478912637998394, 8117321591224225589798160, 32315953861180945578481290, 128687247007646097382575600, 512581888492457685489678720, 2042186800674733339029412800, 8138187367734804408300247080, 32438125142992612042425708750, 129322988328097905200284610484] Number of Good walks sequence [1, 2, 5, 14, 42, 132, 429, 1431, 4873, 16874, 59241, 210392, 754528, 2728704, 9940002, 36439254, 134331805, 497671020, 1851956757, 6919102190, 25943735040, 97596775740, 368242527510, 1393217624799, 5284388988144, 20089855902606, 76540272598795, 292191358965886, 1117503637686624, 4281350850592096, 16429121193273745, 63140123849911920, 243004369791693342, 936491470461819486, 3613610461429817471, 13960342687554963804, 53993318140880202110, 209048633984976937396, 810203670993690181523, 3143102964704314980746, 12204503883231428155605, 47430731825090991581358, 184484581618732399768140, 718132308026243235077362, 2797548847992306464838600, 10905997627499099691269760, 42545558347390277896446100, 166085456484383763434698920, 648762502859852240848514175, 2535744869178390298044796284] {[x^2,0],[0,1]} x squared Number of walks sequence [2, 6, 20, 75, 294, 1176, 4752, 19350, 79320, 327030, 1354860, 5635721, 23521750, 98454480, 413119584, 1737215465, 7319094462, 30888287650, 130552431660, 552540870693, 2341398826954, 9932738455560, 42179628006000, 179283172764150, 762687713313266, 3247099574915700, 13834406451938996, 58981894825983410, 251622662747241390, 1074076639979295200, 4587316497730313792, 19602210743114660100, 83803217435619106950, 358437308458936052080, 1533738280577006529000, 6565449320748290230944, 28115334554814546775528, 120442256409428684947320, 516134910166123443890960, 2212528233782965918775830, 9487432646441352049215384, 40694568641814078665459502, 174600734311762295771291200, 749328032085921683025644475, 3216688373409246304519827548, 13811862128793264259924710832, 59319457039455758593605941280, 254823528581618245724718411175, 1094902150246227340750228215750, 4705445626827856929211114038936] Number of Good walks sequence [1, 2, 5, 15, 49, 168, 594, 2150, 7932, 29730, 112905, 433517, 1680125, 6563632, 25819974, 102189145, 406616359, 1625699350, 6527621583, 26311470033, 106427219407, 431858193720, 1757484500250, 7171326910566, 29334142819741, 120262947219100, 494085944712107, 2033858442275290, 8387422091574713, 34647633547719200, 143353640554072306, 594006386154989700, 2464800512812326675, 10241065955969601488, 42603841127139070250, 177444576236440276512, 739877225126698599356, 3088262984857145767880, 12903372754153086097274, 53964103262999168750630, 225891253486698858314652, 946385317251490201522314, 3968198507085506722074800, 16651734046353815178347655, 69928008117592310967822338, 293869406995601367232440656, 1235822021655328304033457110, 5200480175135066239279967575, 21898043004924546815004564315, 92263639741722684886492432136] {[i,0],[0,1], i = 1..n} Number of walks sequence [2, 9, 44, 225, 1182, 6321, 34232, 187137, 1030490, 5707449, 31760676, 177435297, 994551222, 5590402785, 31500824304, 177880832001, 1006362234162, 5703029112297, 32367243171740, 183945502869345, 1046646207221582, 5961966567317649, 33995080211156904, 194018345610938625, 1108249253656627082, 6335362996269804441, 36242624473448929172, 207471888581918732577, 1188420477299897019750, 6811363130542022605761, 39060208147938813639648, 224107712634529402634241, 1286433270742539835307106, 7387774236621121869452745, 42444810629942707737335052, 243955101254443359534621153, 1402684676069260991029357566, 8068008072375902727223357041, 46421660507252179844474537880, 267187202434949055228909391617, 1538311812510127183691271036858, 8859321018481518005528233587321, 51036148606025306541727392515076, 294083863934074085646717944253345, 1695021356537799691928989850793750, 9772034601190484669564581254991521, 56350287651526935199242378488497872, 325015713959563256579544798068443137, 1875019039169095678630623529660290450, 10819230352459141875641043555829364649] Number of Good walks sequence [1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723, 13648869, 71039373, 372693519, 1968801519, 10463578353, 55909013009, 300159426963, 1618362158587, 8759309660445, 47574827600981, 259215937709463, 1416461675464871, 7760733824437545, 42624971294485657, 234643073935918683, 1294379445480318899, 7154203054548921813, 39614015909996567325, 219721391307807180831, 1220631504623087926239, 6791142807106951594977, 37836272668898230450209, 211079263903460624841507, 1179022517498408548259307, 6593381114984955663097869, 36912754633401605027088357, 206872001855792377621111719, 1160541512681304496111863447, 6516761034998757444607546137, 36626471726431599611696929449, 206030721360035302454144967147, 1159912468318756966857440738979, 6535196976312757458815954316741, 36848290359517384607151953278125, 207915629812563503607757047978543, 1173964326073477816650882885177039, 6632973754276801154684587715682513, 37500380783381913572612470593205809, 212141771616845919130216540310379699] {[0, 1], [1, 0], [1, 2], [2, 1]} Number of walks sequence [2, 10, 46, 226, 1136, 5810, 30080, 157162, 826992, 4376408, 23267332, 124179570, 664919780, 3570265000, 19216805476, 103652442922, 560127574340, 3031887311256, 16435458039076, 89213101943000, 484839755040768, 2637805800869740, 14365506336197816, 78306226704845666, 427205647156264460, 2332463114302134388, 12743966661052722620, 69676148714415162456, 381182773382495626536, 2086574865341293733324, 11427993055862508871096, 62621937187374419689370, 343312923930075146697500, 1882993009451736547314740, 10332156862755093019129420, 56716152796112772561354440, 311448770055353464140154440, 1710889253629332733683604620, 9401661386542605383848339160, 51680576419865197425872650312, 284173305506956836042557253944, 1563025402964060038355575342880, 8599430810230821539614928036432, 47324799125817778709421593851372, 260506229104723560864341521867320, 1434342610142938579277247564278936, 7899300664638801741888520363489480, 43513210548095746230996489089566722, 239743253009776183015534443676916372, 1321176139194582885951056330433131116] Number of Good walks sequence [1, 4, 14, 56, 237, 1046, 4762, 22198, 105430, 508384, 2482297, 12248416, 60980875, 305955356, 1545397464, 7852100294, 40105277640, 205798130604, 1060467961508, 5485199090812, 28469067353686, 148220323891460, 773892318396689, 4051261817405008, 21259240700033317, 111808196895184996, 589247081928527175, 3111401249645431554, 16458592692891091191, 87208007074718248220, 462807919325646496244, 2459710518117877740314, 13090849039943314260430, 69761749022971504972336, 372220866035009916030478, 1988330553379666886745528, 10632956233207494032661636, 56920956781207522174297500, 305014247985027379393823888, 1635974686929801552265294256, 8782574362585022894589703616, 47188652268167198584216282456, 253750401523988129963452758058, 1365568290809039763332167802000, 7354324278568522350347704711466, 39635185789060004022416644760224, 213754258154243276721288488814069, 1153538768046938522869339300802688, 6229048149673391140843752132093589, 33656772440017550895564402952843404] {[0,i],[i,0], i = 1..n} Chess Rook Number of walks sequence [2, 14, 106, 838, 6802, 56190, 470010, 3968310, 33747490, 288654574, 2480593546, 21400729382, 185239360178, 1607913963614, 13991107041306, 122002082809110, 1065855419418690, 9327252391907790, 81744134786314410, 717367363052796678, 6303080714967178962, 55442415802430586174, 488166038461550194746, 4302218944630062062838, 37947642101127770120802, 334977761601783084811950, 2959107762596925411599050, 26157526097740725747576550, 231368076739562525423902450, 2047683932826690652874600990, 18132543469364133527040005210, 160648249609910357850450605270, 1423971310474495839075813171330, 12627638876789041976187994850190, 112028236982985862160090106188010, 994276066546654121109954754622150, 8827772799279501446043112087384850, 78406357404458358041592067899588350, 696625459003895782216384557649814650, 6191386415380254493575775712309193910, 55044014041542814532845424586288521890, 489507683420519847593112536245050910830, 4354418990441188763182784969664212130570, 38745089489057259564343912230309973190310, 344836879640904071004821316599946177821490, 3069845389550082031493667141155930768425950, 27335092010567487528853381282412205841105050, 243456617119484481825794261071350629803238550, 2168780791129804938346519226042313097020000450, 19324089044450126867888747537641583957057623374] Number of Good walks sequence [1, 5, 29, 185, 1257, 8925, 65445, 491825, 3768209, 29324405, 231153133, 1841801065, 14810069497, 120029657805, 979470140661, 8040831465825, 66361595715105, 550284185213925, 4582462506008253, 38306388126997785, 321327658068506121, 2703925940081270205, 22818850380972422469, 193082100135697111185, 1637754327091711568817, 13923022954780587927125, 118610941407900739203725, 1012414535026001576071625, 8657189457343018985130265, 74152981519020587207850925, 636160403591710973227360405, 5465734003674345394560569025, 47025729370355143873580922945, 405128227028878906400461865925, 3494516575054659582714050267485, 30177949910096410986620176053625, 260899447264288743198985221718825, 2257939409057360039486013361620125, 19560678913701886592921326634784485, 169615924219868977334139206123555825, 1472110738341370800624071257886446545, 12787540085872466288806964135311516725, 111170356228359826075288124166983996205, 967231440152816268568473368210895222825, 8421631804513651887431177060896207991865, 73379123845812688647405746997792768682125, 639802743905726483471542617589805691676725, 5582190736388849521407280899960642802286625, 48734518929529394307481374184808479030595169, 425727068078373866450104605347003910383466405] {[0,1],[1,0],[1,1]} Chess King going in NE direction Number of walks sequence [3, 13, 63, 321, 1683, 8989, 48639, 265729, 1462563, 8097453, 45046719, 251595969, 1409933619, 7923848253, 44642381823, 252055236609, 1425834724419, 8079317057869, 45849429914943, 260543813797441, 1482376214227923, 8443414161166173, 48141245001931263, 274738209148561921, 1569245074591690083, 8970232353223635949, 51313576749006450879, 293733710358893793729, 1682471873186160624243, 9642641465118083682429, 55294491352291112747007, 317241780630136241094657, 1820991621200098527441027, 10457362855894862001750669, 60078868458555230983696959, 345299757825442889707393857, 1985346154034162284608274707, 11419126147845924397833900957, 65701922725618214591910684159, 378150244155138145169182750209, 2177132498065215626644755632547, 12538138857330429216252563327277, 72227577760477218461054044462527, 416187677832257853682641940491969, 2398757701327818873799533781222323, 13828965368696585850685952933729469, 79743371991601442215885155629240319, 459935409276618156178021449301004289, 2653337188651000290233505189314055363, 15310086199495855930932559804210504653] Number of Good walks sequence [2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038, 3937603038, 20927156706, 111818026018, 600318853926, 3236724317174, 17518619320890, 95149655201962, 518431875418926, 2832923350929742, 15521467648875090, 85249942588971314, 469286147871837366, 2588758890960637798, 14308406109097843626, 79228031819993134650, 439442782615614361662, 2441263009246175852478, 13582285614213903189954, 75672545337796460900418, 422158527806921249683014, 2358045034996817096518614, 13186762229969911326195738, 73825509266803210054176714, 413744003711584755242223438, 2321083025362608992223726894, 13033522069997514889215092274, 73252943452863199223393858898, 412061442720070604908289934294, 2319824936637513933714881477958, 13070393952625514917631908633482, 73696580719034769214303906556250, 415831259625127007215514095957086, 2347928652146955633301765770354078, 13265947508553602309369175431365026, 75000761566763827145224941186411618, 424283543233691838260433080620759398] {[0, 1], [1, 0], [1, 1], [2, 2]} Number of walks sequence [3, 14, 71, 379, 2082, 11651, 66051, 378064, 2180037, 12644861, 73695358, 431209313, 2531556197, 14904832196, 87970766447, 520337606401, 3083584244460, 18304476242735, 108820740004749, 647817646760368, 3861215365595659, 23039691494489015, 137615812845579390, 822738993163308691, 4922949521986023963, 29480139394909356044, 176664398792213185409, 1059402929496079389631, 6356902648489583020308, 38166637970127629445209, 229276573792993574603787, 1378027658578343598968422, 8286384291210047832045023, 49850407222197537838691609, 300025068023888975708739560, 1806430285705003018542769435, 10880533315007764522316552713, 65559512416370766384694103782, 395157036800184178672318869369, 2382564832244243056285491057263, 14369902278711703328378192730124, 86694302977232341546334036442345, 523177414774768811839762850141075, 3158082974263429490577143873803996, 19068170004920481883275459027866553, 115159713172591157259312917340494129, 695654622575668901314971609202281094, 4203238798456327056130071863122224221, 25401990520261139959363691340584083073, 153546874420395886694107811666429933504] Number of Good walks sequence [2, 7, 27, 116, 532, 2554, 12675, 64507, 334836, 1765833, 9434779, 50962640, 277839361, 1526834471, 8448751385, 47035469902, 263260232668, 1480527858436, 8361881707770, 47409359120684, 269736194796537, 1539547726712437, 8812663513352489, 50579825742416942, 291012706492224315, 1678146650028389023, 9697482307486861125, 56148049483650883418, 325685956663306169704, 1892343209238706561552, 11012606961635116334631, 64184027732131247229285, 374603199042488986947084, 2189213854778309191457243, 12809866709764967142650533, 75043134255268720569687608, 440108608055424139245905601, 2583842686807144572684013803, 15184702028756125034970193936, 89322096703094945357683861273, 525900970632691178245639576571, 3099009333024135237688130129612, 18276723986374036147362693483644, 107873373600843742803060399741842, 637170349334196782845560481892345, 3766241550748448165593651635400583, 22277113593086557880971323751209685, 131854648351194241164905076821344268, 780919270651675579157576052225866419, 4627858707264004716379087916981398517] {[0, 1], [1, 0], [1, 1], [1, 2], [2, 1]} Number of walks sequence [3, 17, 101, 627, 3999, 25955, 170571, 1131433, 7559301, 50795985, 342935689, 2324278669, 15804931797, 107775401349, 736723618773, 5046774983235, 34636814325087, 238114193665451, 1639378334244867, 11301978856210543, 78010917772099207, 539055832175992119, 3728616907056361767, 25814331478115444539, 178871424635719295331, 1240393862877953016659, 8607788344150489508963, 59774281710005361944019, 415343862509689229931603, 2887711024085974599218379, 20087957863759636214580459, 139810289990731556832429705, 973532142385140754360343365, 6781987415575324528312227585, 47265903536300391586112526041, 329543661269363537260560866373, 2298489506004054308723878989597, 16037178132525716652235012259261, 111933792471818551881433942899437, 781510645748128545137587004458569, 5458107876923012054299714098347265, 38130849787237559509045442369471121, 266460027412910059471245190508589633, 1862530256359203249623832369833448561, 13022222994534731140472903742099207729, 91069430285665839468836773086981499161, 637032071502092077244027067819861080697, 4457044092063828399831728599381411955421, 31190730583052112485798063348856952795557, 218319946572315528248192979507107846612901] Number of Good walks sequence [2, 8, 37, 187, 1005, 5633, 32566, 192792, 1162918, 7121902, 44165560, 276783968, 1750207940, 11153013440, 71550898197, 461746445731, 2995462031745, 19523143468973, 127777132138891, 839452832023207, 5533837097342891, 36594085887854795, 242680221398481381, 1613598426851118817, 10754859380597438069, 71842748551816723277, 480905190308830819225, 3225307676284611819217, 21670117616522081549953, 145840679672699944094433, 983053428271497814915238, 6636125814739273692660888, 44859345005089675067344174, 303638741080314921979131686, 2057763386337603966845020668, 13961702500509954172092227612, 94832904877320251327289447656, 644811242965640714216092401476, 4388701626430898296019153105026, 29898414203468535846138390257098, 203868052178975450132547578090878, 1391299997423028635726671660812258, 9502676021281003270491972199172678, 64954415289919470625783673910590162, 444318286253185235104169038220769834, 3041505111745155334565491437675951478, 20834291953518436333856265143938254112, 142808006475300877885644385533683789544, 979486623322036695091659490280293252388, 6722113044436452619618838139392108932544] {[0,i],[i,0],[i,i], i = 1..n} Chess Queen Number of walks sequence [3, 22, 188, 1712, 16098, 154352, 1499858, 14717692, 145509218, 1447187732, 14462966928, 145120265472, 1461040916988, 14751839744412, 149316973768398, 1514654852648052, 15393833895932658, 156716528008129892, 1597861126366223768, 16313712879567298432, 166762790669149883228, 1706588993010917654672, 17482339853119385398788, 179256038008897034917112, 1839581152619511929216948, 18893227697423151203471752, 194182365208362733063474208, 1997132981801013772009084592, 20553097791799261211580209568, 211642219089056982406858914132, 2180546946799680207234558358278, 22477689525456341074573220886372, 231818099272131688444523040664338, 2391883525221999158487840517519812, 24689845516317673342739797454674248, 254959955181369150394859952844937952, 2633851473401486205442283951388374508, 27218707688555819294724918413367081792, 281379547428395639397368136159226247868, 2909771325440517878728896182984495909832, 30099519072775275324517700899035354360828, 311449883420053575804804423737651807898072, 3223581958797604877646832490606418387369888, 33373796073218769640191981063634221844552512, 345607884062282832405108278869822141679859048, 3579875739566095468709594131262627030772484152, 37089757142695004371692040058403900585069727308, 384359383672550371618539588489396080526189433992, 3983955196586923536227924077951030044816307963668, 41302952800706101370724274478545040461255659973032] Number of Good walks sequence [2, 10, 63, 454, 3539, 29008, 246255, 2145722, 19078536, 172402396, 1578687082, 14616730080, 136606848093, 1287022395324, 12210382758519, 116553763025178, 1118580919711060, 10786838228669692, 104469304517331666, 1015700422725526916, 9909822870539332662, 96995587382116080400, 952147466163400968998, 9371687658580682305444, 92469912955030870572624, 914472049133268933537848, 9062647458842054774571240, 89989007808038163040161412, 895190187641540189549803921, 8920314681516160849941020436, 89030054701001664707845727367, 889901053188882780712437650970, 8907509988853235584384765544028, 89278064481332239377057174549564, 895930749444734154618627844533354, 9001511252632307684977622635823268, 90539944218975652593648873397336170, 911640231291066691954058704837189632, 9188440517770771368911777135342223426, 92698679190808404521158445260757482956, 936049080336573114218149698646263693072, 9460157890163266375961569513178804948040, 95687561243152819237226865229860676801308, 968620710759910537567383808866247579415184, 9812464068589417283509455252572151337368834, 99475092878288109510603446586475606336637688, 1009135191000470499868751697691886910076843350, 10244026086640919061672713293352958399327666852, 104055894024005158121054330199275134629242662056, 1057612154591721891480919388648856946832010701976]