---------------------------------------------------------------------------------- A169770 = open knight's tour diagrams of a 3xn chessboard that have "type X": both endpoints occur in the same column 4,0,0,0,80,40,368,352,5296,3744,48656,40208,523808,415488, 5270976,4333504,54215264,44497728,551297184,458337984,5613555008, 4691821600,56981627840,47988689152,577641089664,489273948160, 5845628996352 Asymptotic value .000169*n*3.11949^n when n is even, .0000526*n*3.11949^n when n is odd Generating function: (4*z^4 - 48*z^6 - 288*z^8 + 40*z^9 + 4080*z^10 - 128*z^11 + 8304*z^12 - 4160*z^13 - 129840*z^14 + 9616*z^15 - 56928*z^16 + 147520*z^17 + 2069888*z^18 - 422784*z^19 - 1976096*z^20 - 2638272*z^21 - 16989728*z^22 + 10588096*z^23 + 49127744*z^24 + 25264416*z^25 + 43640128*z^26 - 154038272*z^27 - 480240256*z^28 - 88352640*z^29 + 492078848*z^30 + 1345343232*z^31 + 2283929344*z^32 - 759748352*z^33 - 5889424640*z^34 - 6979236096*z^35 - 3197127424*z^36 + 11457284736*z^37 + 29767983104*z^38 + 18716782592*z^39 - 20244777472*z^40 - 67039148032*z^41 - 79128546304*z^42 - 2928301056*z^43 + 116776381440*z^44 + 215361519616*z^45 + 85158279168*z^46 - 142089760768*z^47 - 253961678848*z^48 - 384278726656*z^49 + 69981736960*z^50 + 415817261056*z^51 + 222768013312*z^52 + 319373246464*z^53 - 211960954880*z^54 - 386769567744*z^55 + 8300527616*z^56 + 8727314432*z^57 - 145041620992*z^58 - 432856956928*z^59 + 131836280832*z^60 - 77447692288*z^61 + 769337458688*z^62 + 1294827388928*z^63 - 907254562816*z^64 - 314502021120*z^65 - 607814680576*z^66 - 752482779136*z^67 + 764349513728*z^68 + 462215970816*z^69 + 509198991360*z^70 - 578335801344*z^71 + 298533781504*z^72 + 121089556480*z^73 - 933282971648*z^74 + 571096432640*z^75 - 757256421376*z^76 - 335846309888*z^77 + 1132394971136*z^78 + 80396419072*z^79 - 66571993088*z^80 - 62545461248*z^81 - 140660178944*z^82 + 24696061952*z^83 - 8589934592*z^84)/ (-1 + 6*z^2 + 64*z^4 - 200*z^6 - 1000*z^8 + 3016*z^10 + 3488*z^12 - 24256*z^14 + 23776*z^16 + 104168*z^18 - 203408*z^20 - 184704*z^22 + 443392*z^24 + 14336*z^26 - 151296*z^28 + 145920*z^30 - 263424*z^32 + 317440*z^34 + 36864*z^36 - 966656*z^38 + 573440*z^40 + 131072*z^42)^2 ---------------------------------------------------------------------------------- A169771 = open knight's tour diagrams of a 3xn chessboard that have "type F": the endpoints occur in different columns and agree in color with the cells in the nearest corner 2, 0, 0, 52, 224, 520, 1616, 10320, 37024, 125120, 441200, 1798576, 6327472, 22985504, 81178008, 301420176, 1057619944, 3818476576, 13412523392, 48285742208, 168992600680, 602349395456, 2106360581920, 7471875943776, 26073917403304, 92017860990176, 320713651212384 Asymptotic value: .02789*3.45059^n Generating function: (2*z^4 - 6*z^5 - 24*z^6 + 106*z^7 - 144*z^8 + 114*z^9 + 1664*z^10 - 6926*z^11 + 7366*z^12 - 3292*z^13 - 31736*z^14 + 127060*z^15 - 305992*z^16 + 725624*z^17 + 558488*z^18 - 157912*z^19 + 7578208*z^20 - 40838744*z^21 - 32722040*z^22 + 21515280*z^23 - 30817528*z^24 + 1135468760*z^25 + 957921304*z^26 - 2127349664*z^27 - 3282178352*z^28 - 18289474624*z^29 - 9766563712*z^30 + 57389285920*z^31 + 88314622656*z^32 + 165936975168*z^33 - 71992550272*z^34 - 805406372704*z^35 - 1003528928032*z^36 - 540007109760*z^37 + 2935329323328*z^38 + 6623410620416*z^39 + 4357767275520*z^40 - 5032135897856*z^41 - 33132735926784*z^42 - 30873041806848*z^43 + 21214374942336*z^44 + 70184194934272*z^45 + 182284323114752*z^46 + 56990857545344*z^47 - 387575520256640*z^48 - 375140079438336*z^49 - 339504370572288*z^50 + 177240746451456*z^51 + 2208204967931392*z^52 + 966073807826944*z^53 - 1810468781539328*z^54 - 1463547344658944*z^55 - 5555536246382080*z^56 - 482260312324608*z^57 + 13527952206238208*z^58 + 4619972116105216*z^59 - 1270892348725248*z^60 - 3418776026994688*z^61 - 33255707248087040*z^62 - 5598351308677120*z^63 + 47530290255192064*z^64 - 1771212246018048*z^65 - 7143687273232384*z^66 - 22854261970313216*z^67 - 113166153151045632*z^68 + 81529112611176448*z^69 + 321446313929924608*z^70 + 118905934343323648*z^71 - 67578776451350528*z^72 - 310285028137697280*z^73 - 1131306052220059648*z^74 - 126851756490555392*z^75 + 1175846449370398720*z^76 + 601363823930769408*z^77 + 1607868619136434176*z^78 - 552787248540811264*z^79 - 3575412627367657472*z^80 - 1015359412059897856*z^81 + 1058337940844576768*z^82 + 2037075530288988160*z^83 + 5087121315676028928*z^84 + 2197803268855037952*z^85 - 7051054534228705280*z^86 - 1304881453988839424*z^87 - 1627839796567605248*z^88 - 4471248508938092544*z^89 + 7805744798075191296*z^90 - 5169887173316444160*z^91 - 4657574446100381696*z^92 + 7833147106725462016*z^93 + 2530012307426115584*z^94 + 10344290678678224896*z^95 + 3603467720181415936*z^96 - 11859357833558491136*z^97 - 10525045902647754752*z^98 - 1904470994244861952*z^99 + 2522461193461825536*z^100 + 11023000479775850496*z^101 + 7186997462064693248*z^102 - 11658174056913436672*z^103 + 883168908167610368*z^104 + 1278871538673647616*z^105 - 5274890571914149888*z^106 + 9721262549378269184*z^107 - 5878821800556625920*z^108 - 12521937887433850880*z^109 + 6026605718558212096*z^110 + 1816199633836179456*z^111 - 3291755966793515008*z^112 + 6126696377306054656*z^113 + 2753824204313853952*z^114 - 2641340110343241728*z^115 + 7097354463601491968*z^116 + 2295013902011858944*z^117 - 7168679714175844352*z^118 - 1839582321632608256*z^119 - 764657423121121280*z^120 - 131219565949485056*z^121 + 1841873841303846912*z^122 + 62435218027446272*z^123 - 372476606339350528*z^124 - 268280837177344*z^125 - 77865214455840768*z^126 - 35223954507431936*z^127 + 53102013575069696*z^128 + 43100855808819200*z^129 + 19333812462813184*z^130 - 9359042975629312*z^131 - 7283165022388224*z^132 + 3448068464705536*z^133 - 703687441776640*z^134 + 281474976710656*z^135)/ ((-1 + 6*z^2 + 64*z^4 - 200*z^6 - 1000*z^8 + 3016*z^10 + 3488*z^12 - 24256*z^14 + 23776*z^16 + 104168*z^18 - 203408*z^20 - 184704*z^22 + 443392*z^24 + 14336*z^26 - 151296*z^28 + 145920*z^30 - 263424*z^32 + 317440*z^34 + 36864*z^36 - 966656*z^38 + 573440*z^40 + 131072*z^42)^2* (1 - 3*z - 9*z^3 - 14*z^4 + 61*z^5 + 110*z^6 + 61*z^7 - 35*z^8 - 498*z^9 - 1262*z^10 - 356*z^11 + 3864*z^12 + 3788*z^13 - 6008*z^14 - 2472*z^15 - 7532*z^16 - 17956*z^17 + 17732*z^18 + 36088*z^19 - 35176*z^20 + 2256*z^21 + 154624*z^22 + 109008*z^23 + 73376*z^24 + 71440*z^25 - 104336*z^26 - 762592*z^27 - 277728*z^28 - 123008*z^29 - 573760*z^30 - 771456*z^31 + 221568*z^32 + 122368*z^33 - 1205760*z^34 + 1966592*z^35 + 714752*z^36 - 2555904*z^37 - 412672*z^38 + 1376256*z^39 - 272384*z^40 - 294912*z^41 + 1781760*z^42 + 860160*z^43 + 516096*z^44 + 1081344*z^45 - 65536*z^46 - 262144*z^47 + 131072*z^48)) ---------------------------------------------------------------------------------- A169772 = open knight's tour diagrams of a 3xn chessboard that have "type B": the endpoints occur in different columns and disagree in color with the cells in the nearest corner 2, 0, 0, 0, 92, 0, 1064, 0, 14928, 0, 156416, 0, 1785600, 0, 19416704, 0, 211014544, 0, 2261999424, 0, 24067157192, 0, 254242274472, 0, 2669251156032, 0, 27880294589248 A169772[n]=0 unless n mod 2 = 0. Asymptotic value: .00144*n*3.11949^n when n is even Generating function: (2*z^4 - 32*z^6 - 48*z^8 + 3296*z^10 - 5886*z^12 - 164032*z^14 + 391524*z^16 + 5212824*z^18 - 12065416*z^20 - 109770584*z^22 + 266379128*z^24 + 1486254456*z^26 - 4708482432*z^28 - 11565226832*z^30 + 62547201600*z^32 + 20674952288*z^34 - 565499322144*z^36 + 617422025024*z^38 + 3154884685888*z^40 - 7903879847296*z^42 - 8916228743296*z^44 + 49649082204928*z^46 + 2800961043840*z^48 - 197892177039360*z^50 + 11451254938624*z^52 + 774942686621696*z^54 + 289061664938496*z^56 - 4840897680566784*z^58 + 165373451073536*z^60 + 27361482729664512*z^62 - 22601140087775232*z^64 - 90168221769553920*z^66 + 159142948298096640*z^68 + 118800993057349632*z^70 - 533938218121052160*z^72 + 181529292851789824*z^74 + 890989951016534016*z^76 - 907838493809704960*z^78 - 309783234974711808*z^80 + 1227863484369469440*z^82 - 1482454816516472832*z^84 - 416526075033812992*z^86 + 2494726588733063168*z^88 - 188921545786130432*z^90 - 406941817722896384*z^92 - 663413869309329408*z^94 - 2920555193030410240*z^96 + 877457406215323648*z^98 + 3217626499423141888*z^100 - 274148839198294016*z^102 - 505635984335962112*z^104 + 1027270910828085248*z^106 - 519974870361047040*z^108 - 2011779264315129856*z^110 + 326857439406194688*z^112 + 30529744231464960*z^114 + 273065293306134528*z^116 + 683954506592944128*z^118 - 42996951959994368*z^120 - 163260984050319360*z^122 - 64457769666740224*z^124 - 6852156464300032*z^126 - 774056185954304*z^128 - 140737488355328*z^130)/((1 - 4*z^2 - 26*z^4 + 4*z^6 - 43*z^8 - 116*z^10 + 888*z^12 + 1224*z^14 + 10292*z^16 + 6052*z^18 - 7088*z^20 + 111280*z^22 - 16192*z^24 - 204080*z^26 + 407232*z^28 - 681472*z^30 + 66432*z^32 - 699392*z^34 - 943104*z^36 - 126976*z^38 + 98304*z^40)* (-1 + 6*z^2 + 64*z^4 - 200*z^6 - 1000*z^8 + 3016*z^10 + 3488*z^12 - 24256*z^14 + 23776*z^16 + 104168*z^18 - 203408*z^20 - 184704*z^22 + 443392*z^24 + 14336*z^26 - 151296*z^28 + 145920*z^30 - 263424*z^32 + 317440*z^34 + 36864*z^36 - 966656*z^38 + 573440*z^40 + 131072*z^42)^2) --------------------------------------------------------------------------------- A169696 = open knight's tour diagrams of a 3xn chessboard 8, 0, 0, 52, 396, 560, 3048, 10672, 57248, 128864, 646272, 1838784, 8636880, 23400992, 105865688, 305753680, 1322849752, 3862974304, 16225820000, 48744080192, 198673312880, 607041217056, 2417584484232, 7519864632928, 29320809649000, 92507134938336, 354439574797984 A169696[n] = A169770[n] + A169771[n] + A169772[n] Asymptotic value: .02789*3.45059^n Generating function: (-4*z^4*(-2 + 6*z + 32*z^2 - 91*z^3 + 108*z^4 - 465*z^5 - 3637*z^6 + 10771*z^7 + 2325*z^8 + 18721*z^9 + 171611*z^10 - 585523*z^11 - 396395*z^12 - 682849*z^13 - 4235681*z^14 + 19633735*z^15 + 18996755*z^16 + 19093765*z^17 + 53067037*z^18 - 478040182*z^19 - 548808389*z^20 - 272099578*z^21 + 99124246*z^22 + 9238848362*z^23 + 10617083638*z^24 - 1326271784*z^25 - 21368020616*z^26 - 141238017880*z^27 - 134392393352*z^28 + 146632364076*z^29 + 549061732140*z^30 + 1593635456720*z^31 + 847737947424*z^32 - 3359839026764*z^33 - 8199885942532*z^34 - 11817729854520*z^35 + 4470847928448*z^36 + 44985860446344*z^37 + 75417736772776*z^38 + 41052444790080*z^39 - 161959393818960*z^40 - 395240529888896*z^41 - 360574399734720*z^42 + 152754263401952*z^43 + 1716838729497632*z^44 + 2369355589689568*z^45 - 116283321402976*z^46 - 2752751355252672*z^47 - 9573026830585088*z^48 - 10836445689890240*z^49 + 11082171458107200*z^50 + 18669207887425408*z^51 + 29684344445342080*z^52 + 52873372588828544*z^53 - 41812705766547456*z^54 - 121892096731426304*z^55 - 129367640322224000*z^56 - 299351435235273344*z^57 - 4272725099376128*z^58 + 959018976196361472*z^59 + 1501626026299465984*z^60 + 1126635861868028416*z^61 - 1312054511814043904*z^62 - 6281405879092401664*z^63 - 9998544859391214592*z^64 + 345842293994065408*z^65 + 23438971031990525440*z^66 + 27613342147451372544*z^67 + 18735021985824688640*z^68 - 27756690114906473472*z^69 - 140273734997306702848*z^70 - 75268328018463164416*z^71 + 138279805585283947520*z^72 + 148601766795439879168*z^73 + 310092565151408475136*z^74 + 99660310022131234816*z^75 - 929030452751789758464*z^76 - 388282869262293155840*z^77 + 606648700399067754496*z^78 + 87564877715883159552*z^79 + 1529476087828639281152*z^80 + 172798188279750090752*z^81 - 5173141273303723016192*z^82 - 277391137321748135936*z^83 + 5234301470080795287552*z^84 + 2982695497894809976832*z^85 + 9153093558466974498816*z^86 - 4247866785647180152832*z^87 - 32038001980536096047104*z^88 - 10236580293112114528256*z^89 + 16318180954295504617472*z^90 + 28945972071705262325760*z^91 + 73306312281463920361472*z^92 - 716787112754376736768*z^93 - 118222134654887364395008*z^94 - 69377035329810834325504*z^95 - 62772131681268743995392*z^96 + 71742019839583955582976*z^97 + 287535417169237064679424*z^98 + 9915871514472405270528*z^99 - 159882598292740527095808*z^100 - 91554524545963890573312*z^101 - 299347948291003811102720*z^102 + 293024518169071175860224*z^103 + 644652542769252884545536*z^104 - 313782304821263993929728*z^105 - 309039150522319333490688*z^106 - 485524416752034730999808*z^107 - 710163496431999149670400*z^108 + 1045217135710526509154304*z^109 + 1343475648128208653516800*z^110 - 572035294664510933041152*z^111 - 624489636735131565686784*z^112 - 651172782170587118174208*z^113 - 1054523143868045305315328*z^114 + 3067547793305302105849856*z^115 + 2309953882806209648525312*z^116 - 1646204301332184416911360*z^117 - 1179316074330817545895936*z^118 - 3361452201934528024936448*z^119 - 1872315354847673124388864*z^120 + 2552418492728380454076416*z^121 + 2312451755176503566925824*z^122 - 2092447391552272559243264*z^123 + 272587570978416396075008*z^124 + 2120420465682278210600960*z^125 - 1464920285536691003850752*z^126 + 6665414743961094974341120*z^127 + 437206599197461019885568*z^128 - 8717997837848029689806848*z^129 + 433562255240785934942208*z^130 - 1051837533156049749540864*z^131 - 1353795424943843357753344*z^132 + 7120052239194963201490944*z^133 + 2694439950503867093876736*z^134 - 5469685596670051324264448*z^135 + 2388057296656668433580032*z^136 + 1857305873625371973255168*z^137 - 6233957703236907652087808*z^138 + 1870622914710554181369856*z^139 - 340193225977415616954368*z^140 - 5678188157950623433621504*z^141 + 2864604435282318749335552*z^142 + 2462256744989383102300160*z^143 - 2068804571445174948855808*z^144 + 1874372066586394439450624*z^145 + 2075481001436645123162112*z^146 - 1165767381159206415499264*z^147 + 1381116855842457274810368*z^148 + 706861976320240724738048*z^149 - 1931359032315239477542912*z^150 + 54548595354419607896064*z^151 + 214962264988336194060288*z^152 - 375280895722266334068736*z^153 + 571140103333652832714752*z^154 + 318446449470697627451392*z^155 - 513105492535911185907712*z^156 - 310588164924270006239232*z^157 - 155527806053471626461184*z^158 - 228675157485332631388160*z^159 + 111435385959294823301120*z^160 + 110547900987225659146240*z^161 + 28404033964839691878400*z^162 + 60151067415078366085120*z^163 + 5503362715849727148032*z^164 + 7434037861704949301248*z^165 + 5203332908674304704512*z^166 - 1944906520677712920576*z^167 + 2346771722627237019648*z^168 - 285348072390194626560*z^169 + 247301662738168676352*z^170 - 19599665578316398592*z^171 + 23058430092136939520*z^172 - 9223372036854775808*z^173 + 4611686018427387904*z^174))/ ((1 - 4*z^2 - 26*z^4 + 4*z^6 - 43*z^8 - 116*z^10 + 888*z^12 + 1224*z^14 + 10292*z^16 + 6052*z^18 - 7088*z^20 + 111280*z^22 - 16192*z^24 - 204080*z^26 + 407232*z^28 - 681472*z^30 + 66432*z^32 - 699392*z^34 - 943104*z^36 - 126976*z^38 + 98304*z^40)* (-1 + 6*z^2 + 64*z^4 - 200*z^6 - 1000*z^8 + 3016*z^10 + 3488*z^12 - 24256*z^14 + 23776*z^16 + 104168*z^18 - 203408*z^20 - 184704*z^22 + 443392*z^24 + 14336*z^26 - 151296*z^28 + 145920*z^30 - 263424*z^32 + 317440*z^34 + 36864*z^36 - 966656*z^38 + 573440*z^40 + 131072*z^42)^2* (1 - 3*z - 9*z^3 - 14*z^4 + 61*z^5 + 110*z^6 + 61*z^7 - 35*z^8 - 498*z^9 - 1262*z^10 - 356*z^11 + 3864*z^12 + 3788*z^13 - 6008*z^14 - 2472*z^15 - 7532*z^16 - 17956*z^17 + 17732*z^18 + 36088*z^19 - 35176*z^20 + 2256*z^21 + 154624*z^22 + 109008*z^23 + 73376*z^24 + 71440*z^25 - 104336*z^26 - 762592*z^27 - 277728*z^28 - 123008*z^29 - 573760*z^30 - 771456*z^31 + 221568*z^32 + 122368*z^33 - 1205760*z^34 + 1966592*z^35 + 714752*z^36 - 2555904*z^37 - 412672*z^38 + 1376256*z^39 - 272384*z^40 - 294912*z^41 + 1781760*z^42 + 860160*z^43 + 516096*z^44 + 1081344*z^45 - 65536*z^46 - 262144*z^47 + 131072*z^48)) ------------------------------------------------------------------------------ A169773 = open knight's tour diagrams of a 3xn chessboard that are symmetric under 180-degree rotation and have "type X": both endpoints occur in the same column 0, 0, 0, 0, 0, 4, 0, 0, 0, 16, 0, 0, 0, 264, 0, 0, 0, 2144, 0, 0, 0, 22408, 0, 0, 0, 211808, 0 A169773[n]=0 unless n mod 4 = 1. Generating function: Xsgf=(2*z*(-2*z^8 + 4*z^12 + 44*z^16 - 168*z^20 + 76*z^24 + 1560*z^28 - 6488*z^32 + 760*z^36 + 38960*z^40 - 45184*z^44 - 47072*z^48 + 128256*z^52 - 109056*z^56 - 174848*z^60 + 357888*z^64 - 74240*z^68 - 294912*z^72 + 176128*z^76 + 49152*z^80))/(-1 + 6*z^4 + 64*z^8 - 200*z^12 - 1000*z^16 + 3016*z^20 + 3488*z^24 - 24256*z^28 + 23776*z^32 + 104168*z^36 - 203408*z^40 - 184704*z^44 + 443392*z^48 + 14336*z^52 - 151296*z^56 + 145920*z^60 - 263424*z^64 + 317440*z^68 + 36864*z^72 - 966656*z^76 + 573440*z^80 + 131072*z^84) ----------------------------------------------------------------------------- A169774 = open knight's tour diagrams of a 3xn chessboard that are symmetric under 180-degree rotation and have "type F": the endpoints occur in different columns and agree in color with the cells in the nearest corner 2, 0, 0, 4, 12, 20, 28, 120, 104, 304, 384, 1304, 1680, 4936, 5908, 18304, 21412, 63440, 76920, 233248, 281284, 833720, 990104, 2993016, 3523740, 10485472, 12432392 Generating function: 2*((-z^4 + 3*z^6 - 5*z^10 + 104*z^12 - 313*z^14 - 4*z^16 - 171*z^18 - 2339*z^20 + 8860*z^22 + 9784*z^24 + 4890*z^26 + 18156*z^28 - 153234*z^30 - 343960*z^32 - 82314*z^34 + 183616*z^36 + 2167892*z^38 + 4674684*z^40 + 1214200*z^42 - 7642448*z^44 - 24839584*z^46 - 25290272*z^48 + 2639080*z^50 + 97909800*z^52 + 198996352*z^54 - 52463592*z^56 - 289294128*z^58 - 511923376*z^60 - 759501216*z^62 + 1292528608*z^64 + 2873753888*z^66 - 44012544*z^68 - 1072188256*z^70 - 3710764672*z^72 - 9800009568*z^74 + 8694927328*z^76 + 18872498816*z^78 - 11899505344*z^80 - 9113655040*z^82 - 19487083008*z^84 - 32500681600*z^86 + 81858293376*z^88 + 129594129280*z^90 - 39014011264*z^92 - 116225499392*z^94 - 171132619008*z^96 - 209781729792*z^98 + 232385627136*z^100 + 408897419776*z^102 + 260178734080*z^104 - 287482004480*z^106 - 213359142912*z^108 - 101815662592*z^110 - 440518531072*z^112 + 1148036644864*z^114 - 579017195520*z^116 - 892111355904*z^118 + 301642235904*z^120 - 664092180480*z^122 + 720292691968*z^124 + 65720205312*z^126 - 70804504576*z^128 - 1618859393024*z^130 + 335592292352*z^132 + 2077301473280*z^134 - 100768677888*z^136 + 2442447814656*z^138 + 661224947712*z^140 - 2300255404032*z^142 + 132761255936*z^144 + 609193295872*z^146 - 1134952448000*z^148 - 503542972416*z^150 + 158337073152*z^152 - 497576574976*z^154 - 138797907968*z^156 + 664746852352*z^158 - 709818843136*z^160 + 366766718976*z^162 + 601328975872*z^164 - 589249380352*z^166 + 251725348864*z^168 - 290312945664*z^170 - 105763569664*z^172 + 127238406144*z^174 - 17716740096*z^176 + 33285996544*z^178 - 10737418240*z^180 - 4294967296*z^182)/((-1 + 6*z^4 + 64*z^8 - 200*z^12 - 1000*z^16 + 3016*z^20 + 3488*z^24 - 24256*z^28 + 23776*z^32 + 104168*z^36 - 203408*z^40 - 184704*z^44 + 443392*z^48 + 14336*z^52 - 151296*z^56 + 145920*z^60 - 263424*z^64 + 317440*z^68 + 36864*z^72 - 966656*z^76 + 573440*z^80 + 131072*z^84)*(1 - 3*z^2 - 9*z^6 - 14*z^8 + 61*z^10 + 110*z^12 + 61*z^14 - 35*z^16 - 498*z^18 - 1262*z^20 - 356*z^22 + 3864*z^24 + 3788*z^26 - 6008*z^28 - 2472*z^30 - 7532*z^32 - 17956*z^34 + 17732*z^36 + 36088*z^38 - 35176*z^40 + 2256*z^42 + 154624*z^44 + 109008*z^46 + 73376*z^48 + 71440*z^50 - 104336*z^52 - 762592*z^54 - 277728*z^56 - 123008*z^58 - 573760*z^60 - 771456*z^62 + 221568*z^64 + 122368*z^66 - 1205760*z^68 + 1966592*z^70 + 714752*z^72 - 2555904*z^74 - 412672*z^76 + 1376256*z^78 - 272384*z^80 - 294912*z^82 + 1781760*z^84 + 860160*z^86 + 516096*z^88 + 1081344*z^90 - 65536*z^92 - 262144*z^94 + 131072*z^96)) + (z*(-2*z^6 - 4*z^8 - 18*z^10 + 70*z^12 + 230*z^14 + 26*z^16 + 1618*z^18 - 5170*z^20 - 7324*z^22 + 282*z^24 - 30928*z^26 + 113600*z^28 + 137148*z^30 - 23704*z^32 + 104224*z^34 - 1213528*z^36 - 1205652*z^38 + 255852*z^40 + 2640760*z^42 + 8379384*z^44 - 2307144*z^46 + 1806328*z^48 - 31681480*z^50 - 41497144*z^52 + 119192128*z^54 - 5490480*z^56 + 107394432*z^58 + 81750080*z^60 - 831208160*z^62 - 268130176*z^64 + 71838656*z^66 + 567812832*z^68 + 2610802496*z^70 + 1660691808*z^72 + 202592256*z^74 - 3817204096*z^76 - 7249009280*z^78 + 4693604224*z^80 - 5689640448*z^82 - 12591207680*z^84 + 23474074624*z^86 - 52146535296*z^88 - 33293224448*z^90 + 130535052032*z^92 + 41046635008*z^94 + 65689009664*z^96 + 218054170624*z^98 - 89538932224*z^100 - 324887615488*z^102 + 230488592384*z^104 - 148101219328*z^106 - 779282309120*z^108 + 82453209088*z^110 - 554472839168*z^112 - 215170654208*z^114 + 1217440272384*z^116 + 436570284032*z^118 + 68253294592*z^120 - 893611925504*z^122 - 146271518720*z^124 + 587693621248*z^126 - 10622173184*z^128 + 356122263552*z^130 - 1536981663744*z^132 - 2070867607552*z^134 - 108955435008*z^136 + 3310107230208*z^138 + 1444671651840*z^140 - 787301793792*z^142 + 1214091100160*z^144 - 1836802637824*z^146 - 1849954402304*z^148 + 1553157062656*z^150 + 548474454016*z^152 - 1530746896384*z^154 + 813808222208*z^156 + 2080601276416*z^158 - 444990488576*z^160 - 482814722048*z^162 - 3657433088*z^164 - 309573189632*z^166 - 15837691904*z^168 - 22414360576*z^170 - 38923141120*z^172 + 82141249536*z^174 - 20937965568*z^176 - 11811160064*z^178 + 10737418240*z^180 - 4294967296*z^182))/((-1 + 6*z^4 + 64*z^8 - 200*z^12 - 1000*z^16 + 3016*z^20 + 3488*z^24 - 24256*z^28 + 23776*z^32 + 104168*z^36 - 203408*z^40 - 184704*z^44 + 443392*z^48 + 14336*z^52 - 151296*z^56 + 145920*z^60 - 263424*z^64 + 317440*z^68 + 36864*z^72 - 966656*z^76 + 573440*z^80 + 131072*z^84)*(1 - 3*z^2 - 9*z^6 - 14*z^8 + 61*z^10 + 110*z^12 + 61*z^14 - 35*z^16 - 498*z^18 - 1262*z^20 - 356*z^22 + 3864*z^24 + 3788*z^26 - 6008*z^28 - 2472*z^30 - 7532*z^32 - 17956*z^34 + 17732*z^36 + 36088*z^38 - 35176*z^40 + 2256*z^42 + 154624*z^44 + 109008*z^46 + 73376*z^48 + 71440*z^50 - 104336*z^52 - 762592*z^54 - 277728*z^56 - 123008*z^58 - 573760*z^60 - 771456*z^62 + 221568*z^64 + 122368*z^66 - 1205760*z^68 + 1966592*z^70 + 714752*z^72 - 2555904*z^74 - 412672*z^76 + 1376256*z^78 - 272384*z^80 - 294912*z^82 + 1781760*z^84 + 860160*z^86 + 516096*z^88 + 1081344*z^90 - 65536*z^92 - 262144*z^94 + 131072*z^96))) --------------------------------------------------------------------------------- A169775 = open knight's tour diagrams of a 3xn chessboard that are symmetric under 180-degree rotation and have "type B": the endpoints occur in different columns and disagree in color with the cells in the nearest corner A169775[n]=0 unless n mod 2 = 0 2, 0, 0, 0, 8, 0, 16, 0, 48, 0, 200, 0, 616, 0, 1832, 0, 6008, 0, 19304, 0, 62180, 0, 189580, 0, 615792, 0, 1895952 Generating function: (2*(-z^4 + 6*z^8 - 8*z^10 + 82*z^12 - 20*z^14 - 420*z^16 + 612*z^18 - 2601*z^20 + 1180*z^22 + 9636*z^24 - 13790*z^26 + 30020*z^28 - 8360*z^30 - 111666*z^32 + 125802*z^34 + 3480*z^36 - 296804*z^38 + 750328*z^40 - 453708*z^42 - 2396064*z^44 + 4422112*z^46 - 1718592*z^48 + 249320*z^50 + 13126336*z^52 - 7351032*z^54 - 11747336*z^56 - 8562480*z^58 + 10574768*z^60 - 127472576*z^62 - 43866112*z^64 + 322119296*z^66 - 63488704*z^68 - 15842816*z^70 + 1932813280*z^72 - 3473201536*z^74 - 3031890848*z^76 + 7671718784*z^78 - 11645743040*z^80 + 13684587392*z^82 + 27803501440*z^84 - 43075975936*z^86 + 14020632832*z^88 + 2223843584*z^90 - 110886256000*z^92 + 106958197760*z^94 + 101207915264*z^96 - 141485612032*z^98 + 228906284032*z^100 - 138821877248*z^102 - 372102619136*z^104 + 243927770112*z^106 - 161133627392*z^108 + 138171949056*z^110 + 388170680320*z^112 + 101717489664*z^114 - 342422921216*z^116 - 15175540736*z^118 - 311896588288*z^120 - 268558786560*z^122 + 804929388544*z^124 - 606094655488*z^126 + 458928259072*z^128 - 273174396928*z^130 - 412299427840*z^132 - 92658991104*z^134 - 540368699392*z^136 + 737550532608*z^138 - 333619396608*z^140 + 344268472320*z^142 + 63964184576*z^144 + 341981528064*z^146 + 972996739072*z^148 - 193533575168*z^150 + 85698019328*z^152 - 39837499392*z^154 - 386060517376*z^156 - 154115506176*z^158 - 106837311488*z^160 - 34695282688*z^162 - 47110422528*z^164 - 7247757312*z^166 - 3758096384*z^168))/ ((1 - 4*z^4 - 26*z^8 + 4*z^12 - 43*z^16 - 116*z^20 + 888*z^24 + 1224*z^28 + 10292*z^32 + 6052*z^36 - 7088*z^40 + 111280*z^44 - 16192*z^48 - 204080*z^52 + 407232*z^56 - 681472*z^60 + 66432*z^64 - 699392*z^68 - 943104*z^72 - 126976*z^76 + 98304*z^80)*(-1 + 6*z^4 + 64*z^8 - 200*z^12 - 1000*z^16 + 3016*z^20 + 3488*z^24 - 24256*z^28 + 23776*z^32 + 104168*z^36 - 203408*z^40 - 184704*z^44 + 443392*z^48 + 14336*z^52 - 151296*z^56 + 145920*z^60 - 263424*z^64 + 317440*z^68 + 36864*z^72 - 966656*z^76 + 573440*z^80 + 131072*z^84)) --------------------------------------------------------------------------------- A169776 = geometrically distinct open knight's tours of a 3xn chessboard that have twofold symmetry 2, 0, 0, 2, 10, 12, 22, 60, 76, 160, 292, 652, 1148, 2600, 3870, 9152, 13710, 32792, 48112, 116624, 171732, 428064, 589842, 1496508, 2069766, 5348640, 7164172 A169776[n]=(A169773[n]+A169774[n]+A169775[n])/2 Generating function: (2*(-z^4 + 6*z^8 - 8*z^10 + 82*z^12 - 20*z^14 - 420*z^16 + 612*z^18 - 2601*z^20 + 1180*z^22 + 9636*z^24 - 13790*z^26 + 30020*z^28 - 8360*z^30 - 111666*z^32 + 125802*z^34 + 3480*z^36 - 296804*z^38 + 750328*z^40 - 453708*z^42 - 2396064*z^44 + 4422112*z^46 - 1718592*z^48 + 249320*z^50 + 13126336*z^52 - 7351032*z^54 - 11747336*z^56 - 8562480*z^58 + 10574768*z^60 - 127472576*z^62 - 43866112*z^64 + 322119296*z^66 - 63488704*z^68 - 15842816*z^70 + 1932813280*z^72 - 3473201536*z^74 - 3031890848*z^76 + 7671718784*z^78 - 11645743040*z^80 + 13684587392*z^82 + 27803501440*z^84 - 43075975936*z^86 + 14020632832*z^88 + 2223843584*z^90 - 110886256000*z^92 + 106958197760*z^94 + 101207915264*z^96 - 141485612032*z^98 + 228906284032*z^100 - 138821877248*z^102 - 372102619136*z^104 + 243927770112*z^106 - 161133627392*z^108 + 138171949056*z^110 + 388170680320*z^112 + 101717489664*z^114 - 342422921216*z^116 - 15175540736*z^118 - 311896588288*z^120 - 268558786560*z^122 + 804929388544*z^124 - 606094655488*z^126 + 458928259072*z^128 - 273174396928*z^130 - 412299427840*z^132 - 92658991104*z^134 - 540368699392*z^136 + 737550532608*z^138 - 333619396608*z^140 + 344268472320*z^142 + 63964184576*z^144 + 341981528064*z^146 + 972996739072*z^148 - 193533575168*z^150 + 85698019328*z^152 - 39837499392*z^154 - 386060517376*z^156 - 154115506176*z^158 - 106837311488*z^160 - 34695282688*z^162 - 47110422528*z^164 - 7247757312*z^166 - 3758096384*z^168))/ ((1 - 4*z^4 - 26*z^8 + 4*z^12 - 43*z^16 - 116*z^20 + 888*z^24 + 1224*z^28 + 10292*z^32 + 6052*z^36 - 7088*z^40 + 111280*z^44 - 16192*z^48 - 204080*z^52 + 407232*z^56 - 681472*z^60 + 66432*z^64 - 699392*z^68 - 943104*z^72 - 126976*z^76 + 98304*z^80)*(-1 + 6*z^4 + 64*z^8 - 200*z^12 - 1000*z^16 + 3016*z^20 + 3488*z^24 - 24256*z^28 + 23776*z^32 + 104168*z^36 - 203408*z^40 - 184704*z^44 + 443392*z^48 + 14336*z^52 - 151296*z^56 + 145920*z^60 - 263424*z^64 + 317440*z^68 + 36864*z^72 - 966656*z^76 + 573440*z^80 + 131072*z^84)) -------------------------------------------------------------------------------- A169777 = geometrically distinct open knight's tours of a 3xn chessboard 3, 0, 0, 14, 104, 146, 773, 2698, 14350, 32296, 161714, 460022, 2159794, 5851548, 26468357, 76442996, 330719293, 965759972, 4056479056, 12186078360, 49668414086, 151760518296, 604396415979, 1879966906486, 7330203447133, 23126786408904, 88609897281582 A169777[n]=A169696[n]/4+A169776[n]/2 The three distinct 3x4 tours were published by Euler in Memoires Acad. Roy. Sci. (Berlin, 1759), 310--337. Generating function: -((z^4*(-2 + 6*z + 32*z^2 - 91*z^3 + 108*z^4 - 465*z^5 - 3637*z^6 + 10771*z^7 + 2325*z^8 + 18721*z^9 + 171611*z^10 - 585523*z^11 - 396395*z^12 - 682849*z^13 - 4235681*z^14 + 19633735*z^15 + 18996755*z^16 + 19093765*z^17 + 53067037*z^18 - 478040182*z^19 - 548808389*z^20 - 272099578*z^21 + 99124246*z^22 + 9238848362*z^23 + 10617083638*z^24 - 1326271784*z^25 - 21368020616*z^26 - 141238017880*z^27 - 134392393352*z^28 + 146632364076*z^29 + 549061732140*z^30 + 1593635456720*z^31 + 847737947424*z^32 - 3359839026764*z^33 - 8199885942532*z^34 - 11817729854520*z^35 + 4470847928448*z^36 + 44985860446344*z^37 + 75417736772776*z^38 + 41052444790080*z^39 - 161959393818960*z^40 - 395240529888896*z^41 - 360574399734720*z^42 + 152754263401952*z^43 + 1716838729497632*z^44 + 2369355589689568*z^45 - 116283321402976*z^46 - 2752751355252672*z^47 - 9573026830585088*z^48 - 10836445689890240*z^49 + 11082171458107200*z^50 + 18669207887425408*z^51 + 29684344445342080*z^52 + 52873372588828544*z^53 - 41812705766547456*z^54 - 121892096731426304*z^55 - 129367640322224000*z^56 - 299351435235273344*z^57 - 4272725099376128*z^58 + 959018976196361472*z^59 + 1501626026299465984*z^60 + 1126635861868028416*z^61 - 1312054511814043904*z^62 - 6281405879092401664*z^63 - 9998544859391214592*z^64 + 345842293994065408*z^65 + 23438971031990525440*z^66 + 27613342147451372544*z^67 + 18735021985824688640*z^68 - 27756690114906473472*z^69 - 140273734997306702848*z^70 - 75268328018463164416*z^71 + 138279805585283947520*z^72 + 148601766795439879168*z^73 + 310092565151408475136*z^74 + 99660310022131234816*z^75 - 929030452751789758464*z^76 - 388282869262293155840*z^77 + 606648700399067754496*z^78 + 87564877715883159552*z^79 + 1529476087828639281152*z^80 + 172798188279750090752*z^81 - 5173141273303723016192*z^82 - 277391137321748135936*z^83 + 5234301470080795287552*z^84 + 2982695497894809976832*z^85 + 9153093558466974498816*z^86 - 4247866785647180152832*z^87 - 32038001980536096047104*z^88 - 10236580293112114528256*z^89 + 16318180954295504617472*z^90 + 28945972071705262325760*z^91 + 73306312281463920361472*z^92 - 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