cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A194646 Number of ways to place 4n nonattacking kings on an 8 X 2n cylindrical chessboard.

Original entry on oeis.org

80, 276, 1082, 4460, 18890, 81606, 358564, 1599820, 7238864, 33175486, 153802520, 720390254, 3404944506, 16221905696, 77820675992, 375564803020, 1821845982082, 8876847931644, 43416046650306, 213033152875350, 1048198981050148, 5169676077206180
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 31 2011

Keywords

Comments

This cylinder is horizontal: a chessboard where it is supposed that rows 1 and 2n are in contact (number of columns = 8, number of rows = 2n).

Links

Crossrefs

Cf. A173782, A194644, A194645, A137432, A195592.

Programs

  • Mathematica
    Table[FullSimplify[4+2*5^n+2*4^n + 2*(2+Sqrt[3])^n + 2*(2-Sqrt[3])^n+ 4*((5+Sqrt[5])/2)^n + 4*((5-Sqrt[5])/2)^n + 4*((5+Sqrt[13])/2)^n+4*((5-Sqrt[13])/2)^n+ 2*(2*Cos[Pi/7])^(2n) + 2*(2*Cos[2*Pi/7])^(2n) + 2*(2*Cos[3*Pi/7])^(2n)], {n,10}]

Formula

a(n) = 4 + 2*5^n + 2*4^n + 2*(2+sqrt(3))^n+2*(2-sqrt(3))^n + 4*((5+sqrt(5))/2)^n+4*((5-sqrt(5))/2)^n + 4*((5+sqrt(13))/2)^n+4*((5-sqrt(13))/2)^n + 2*(2*cos(Pi/7))^(2n)+2*(2*cos(2*Pi/7))^(2n)+2*(2*cos(3*Pi/7))^(2n).
Recurrence: a(n) = -300*a(n-12) + 4235*a(n-11) - 23320*a(n-10) + 66422*a(n-9) - 111545*a(n-8) + 118727*a(n-7) - 83449*a(n-6) + 39539*a(n-5) - 12676*a(n-4) + 2708*a(n-3) - 369*a(n-2) + 29*a(n-1).
G.f.: -2*(-17 + 453*x - 5251*x^2 + 34737*x^3 - 144635*x^4 + 394423*x^5 - 711101*x^6 + 836705*x^7 - 620007*x^8 + 270365*x^9 - 61055*x^10 + 5335*x^11)/((-1+x)*(-1+4*x)*(-1+5*x)*(1-4*x+x^2)*(1-5*x+3*x^2)*(1-5*x+5*x^2)*(-1+5*x-6*x^2+x^3)).

A194647 Number of ways to place 5n nonattacking kings on a 10 X 2n cylindrical chessboard.

Original entry on oeis.org

192, 708, 3036, 13932, 66532, 327192, 1649420, 8500668, 44693472, 239238888, 1301236304, 7177627944, 40078823652, 226167613792, 1287874058656, 7390391650172, 42688584938548, 247956702607932, 1447080255512308, 8479116559291112, 49852445684576540
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 31 2011

Keywords

Comments

This cylinder is horizontal: a chessboard where it is supposed that rows 1 and 2n are in contact (number of columns = 10, number of rows = 2n).

Links

Crossrefs

Cf. A173783, A194644, A194645, A194646, A137432, A195593.

Formula

G.f.: -2*(7089408*x^21 - 132938496*x^20 + 1125112128*x^19 - 5717239392*x^18 + 19578445344*x^17 - 48082847384*x^16 + 88003026752*x^15 - 123138008952*x^14 + 134072006560*x^13 - 114991853490*x^12 + 78336556962*x^11 - 42596878318*x^10 + 18524447581*x^9 - 6435525481*x^8 + 1778018953*x^7 - 387290192*x^6 + 65568715*x^5 - 8436954*x^4 + 796245*x^3 - 51918*x^2 + 2088*x - 39)/((x-1)*(2*x-1)*(4*x-1)*(6*x-1)*(x^2-4*x+1)*(2*x^2-5*x+1)*(2*x^2-4*x+1)*(4*x^2-6*x+1)*(6*x^2-6*x+1)*(7*x^2-6*x+1)*(2*x^3-8*x^2+6*x-1)*(3*x^3-9*x^2+6*x-1)).
Asymptotic: a(n) ~ 2*6^n.

A194648 Number of ways to place 6n nonattacking kings on a 12 X 2n cylindrical chessboard.

Original entry on oeis.org

448, 1732, 7918, 39316, 205628, 1118398, 6286658, 36383284, 216134044, 1314160492, 8155899320, 51526819510, 330559583178, 2148524237842, 14120142260138, 93669254201140, 626289974615094, 4215364545901036, 28531464984810918, 194028126730583796
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 31 2011

Keywords

Comments

This cylinder is horizontal: a chessboard where it is supposed that rows 1 and 2n are in contact (number of columns = 12, number of rows = 2n).

Links

  • Ray Chandler, Table of n, a(n) for n = 1..1182
  • V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights
  • Vaclav Kotesovec, G.f., recurrence and explicit formula
  • Index entries for linear recurrences with constant coefficients, signature (123, -7321, 280953, -7815096, 167948132, -2902322885, 41450235787, -499005760654, 5139688443504, -45815780494341, 356686744049422, -2442977988253415, 14807521518627422, -79813193163925620, 384075945698236159, -1655461203049171846, 6408095608117333736, -22324060511037585983, 70110767534529096739, -198758740577568913437, 509105053427808565774, -1178957762144560119277, 2469102261987638350625, -4676788353519183205994, 8009846446628877143184, -12398045659209581154381, 17330242433059284917247, -21854027730161965985121, 24829391878777653306741, -25375425632156076733736, 23283124318667970185580, -19136737348463193623732, 14052187658858602811086, -9190302655918861018594, 5334155219056391326621, -2736099335631729732358, 1234232572662337908295, -486797186990900726158, 166723091105251092029, -49174878927647152193, 12365157710942649267, -2617610437109862140, 459103033290045078, -65327706620774553, 7328352058508736, -621736878459564, 37366066553760, -1412422401600, 25147584000).

Crossrefs

Cf. A174154, A194644, A194645, A194646, A194647, A137432, A195594.

Formula

Asymptotic: a(n) ~ 2*7^n.

A195004 Number of ways to place 7n nonattacking kings on a 14 X 2n cylindrical chessboard.

Original entry on oeis.org

1024, 4100, 19648, 103508, 580664, 3419648, 20984924, 133538996, 877751236, 5937279840, 41180193352, 291859775552, 2106967145904, 15448890481568, 114765555945488, 861942483797204, 6533144250310688, 49899718750389380, 383593821097441412, 2964842429047018248
Offset: 1

Views

Author

Vaclav Kotesovec, Sep 07 2011

Keywords

Comments

This cylinder is horizontal: a chessboard where it is supposed that rows 1 and 2n are in contact (number of columns = 14, number of rows = 2n).

Links

  • Ray Chandler, Table of n, a(n) for n = 1..1106
  • V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights
  • Vaclav Kotesovec, G.f., recurrence and explicit formula
  • Index entries for linear recurrences with constant coefficients, signature (234, -26876, 2019709, -111697218, 4847894553, -171967643248, 5126864261813, -131103713766739, 2920551204572253, -57369727964797769, 1003499987301575705, -15756273391572396056, 223560731166583964170, -2882674835842994926649, 33942752076287377374732, -366481987107253117428678, 3641474337928260994018833, -33403183417194915064568472, 283651397179433118312259120, -2235256807858321803493302712, 16381469343464643012631705641, -111864817301276143786034674593, 713000590137978010047099019123, -4248167585872182220244662296078, 23692825139854478095829187332583, -123839650151492238580849239743842, 607291338747966869409975798808119, -2796687736808231684915622029956501, 12105097441367664211274616723997585, -49282739018956175502285900368718340, 188846441297378176522518597171831038, -681488797315874299501618146366584502, 2317175918409451996565304916085049225, -7426636629919740985496045649451822465, 22444540561400607122897712881328419731, -63979187901854273527092491251827548655, 172057350932125535425088191059644660790, -436601690854837262203616552560815387262, 1045493932598927036221255467761014916672, -2362673225064751662372547547704555633600, 5038838587531054046651836530047833987628, -10140921814167342121211938118088849001792, 19257391087913181405263225040629151186272, -34500132785018636006710664752520219397318, 58298198399790056471455253509791714097825, -92893574441178572445869535970403724406464, 139532260661304247096543190046679153931782, -197497210344518691909307839617377332496726, 263305424445938364650937820234766261392812, -330492584893065300696048626280270629928420, 390329890315638397879899046383015257779768, -433519474585554675321027469844536795515096, 452483527667626510122157361959823577545116, -443501901490900685964486690632625788788496, 407883969421282557544660851181757217890384, -351675933152655222446863110197897961028664, 283984050472933807032016387053214808490768, -214551731812442749852409765446269884375456, 151480235225815576544104717761138879307344, -99820208079286562792944814404190179497760, 61308554282442277738368818159312699696768, -35043595589343382666629911544555460702048, 18610707623818189757146652551568358579392, -9166308055333364648206086359099895891840, 4178629157017923157800455107231658228096, -1759220243521242971272733964440808344960, 682319955535117483948474380530632894464, -243136873223743567485233920640254062592, 79356174222862273290100936536508525568, -23641977426444551429474710014044615168, 6404251284485474979502046840644695040, -1570379926552348612740435108569370624, 346795140719354500498066916678473728, -68563667046969861210140030991157248, 12051369611781880146558356044689408, -1867616610183259872317123737116672, 252622266308749887603294277730304, -29456297112310112325526951329792, 2914402636658640861712637558784, -239667516215588644643093741568, 15925650465805208711732920320, -820834685159380646929367040, 30762599549292146117836800, -745091435273014738944000, 8746516217730170880000).

Crossrefs

Cf. A174155, A194644, A194645, A194646, A194647, A194648, A137432, A195595.

A195591 Number of ways to place 3n nonattacking kings on a vertical cylinder 6 X 2n.

Original entry on oeis.org

16, 90, 344, 1082, 3036, 7918, 19648, 47058, 109796, 251126, 565512, 1257754, 2769196, 6046014, 13107536, 28246370, 60555636, 129237382, 274727320, 581960106, 1228931516, 2587886030, 5435818464, 11391730162, 23823647236, 49727668758, 103616086568
Offset: 1

Views

Author

Vaclav Kotesovec, Sep 21 2011

Keywords

Comments

Vertical cylinder: a chessboard where it is supposed that the columns 1 and 6 are in contact (number of columns = 6, number of rows = 2n).

Links

Crossrefs

Cf. A194645, A061594, A137432.

Programs

  • Mathematica
    LinearRecurrence[{6,-13,12,-4},{16,90,344,1082},30] (* Harvey P. Dale, Nov 15 2021 *)

Formula

Recurrence: a(n) = -4*a(n-4) + 12*a(n-3) - 13*a(n-2) + 6*a(n-1).
G.f.: x*(1+10*x+7*x^2)/((x-1)^2*(2*x-1)^2).
a(n) = (31*n - 65)*2^n + 18*n + 66.
E.g.f.: exp(x)*(48*(1 - exp(x)) + x*(18 + 31*exp(x))). - Stefano Spezia, Aug 31 2025
Showing 1-5 of 5 results.

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