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 1-4 of 4 results.

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.

A195592 Number of ways to place 4n nonattacking kings on a vertical cylinder 8 X 2n.

Original entry on oeis.org

32, 256, 1220, 4460, 13932, 39316, 103508, 259372, 626780, 1473764, 3392964, 7682812, 17166476, 37942900, 83115188, 180699980, 390351420, 838619524, 1793087780, 3817890076, 8099228012, 17125372436, 36104600340, 75916936300, 159249370652, 333329766436
Offset: 1

Views

Author

Vaclav Kotesovec, Sep 21 2011

Keywords

Comments

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

Links

Crossrefs

Cf. A194646, A173782, A137432.

Programs

  • Mathematica
    LinearRecurrence[{7,-19,25,-16,4},{32,256,1220,4460,13932},30] (* Harvey P. Dale, Jan 15 2016 *)

Formula

Recurrence: a(n) = 4*a(n-5) - 16*a(n-4) + 25*a(n-3) - 19*a(n-2) + 7*a(n-1).
G.f.: -(1 + 25*x + 51*x^2 + 11*x^3)/((x-1)^3*(2*x-1)^2).
a(n) = (221*n - 779)*2^n + 44*n^2 + 324*n + 780.
Showing 1-4 of 4 results.

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