A137432 -id:A137432 - cp's OEIS frontend

A195590 Number of ways to place 2n nonattacking kings on a vertical cylinder 4 X 2n.

8, 32, 100, 276, 708, 1732, 4100, 9476, 21508, 48132, 106500, 233476, 507908, 1097732, 2359300, 5046276, 10747908, 22806532, 48234500, 101711876, 213909508, 448790532, 939524100, 1962934276, 4093640708, 8522825732, 17716740100, 36775657476, 76235669508

Offset: 1

Vaclav Kotesovec, Sep 21 2011

nonn

Vertical cylinder: a chessboard where it is supposed that the columns 1 and 4 are in contact (number of columns = 4, number of rows = 2n).

Ray Chandler, Table of n, a(n) for n = 1..3307
V. Kotesovec, Non-attacking chess pieces
Index entries for linear recurrences with constant coefficients, signature (5, -8, 4).

Cf. A194644, A061593, A137432.

Recurrence: a(n) = 4*a(n-3) - 8*a(n-2) + 5*a(n-1).
G.f.: -(1+3*x)/((x-1)*(2*x-1)^2).
Explicit formula: a(n) = (5*n-3)*2^n + 4.

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

Vaclav Kotesovec, Sep 21 2011

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).

Ray Chandler, Table of n, a(n) for n = 1..3305
V. Kotesovec, Non-attacking chess pieces.
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Cf. A194645, A061594, A137432.

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

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

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

Vaclav Kotesovec, Sep 21 2011

nonn

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).

Ray Chandler, Table of n, a(n) for n = 1..3302
V. Kotesovec, Non-attacking chess pieces
Index entries for linear recurrences with constant coefficients, signature (7, -19, 25, -16, 4).

Cf. A194646, A173782, A137432.

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

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.

A195593 Number of ways to place 5n nonattacking kings on a vertical cylinder 10 X 2n.

Original entry on oeis.org

64, 732, 4392, 18890, 66532, 205628, 580664, 1536814, 3877300, 9434784, 22327496, 51698178, 117645348, 263992580, 585640568, 1286898262, 2805399156, 6074441896, 13076687560, 28009586346, 59732295204, 126891641612, 268638308152, 566987715710

Offset: 1

Vaclav Kotesovec, Sep 21 2011

nonn

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

Ray Chandler, Table of n, a(n) for n = 1..3299 (first 1000 terms from Jinyuan Wang)
Vaclav Kotesovec, Non-attacking chess pieces
Index entries for linear recurrences with constant coefficients, signature (8,-26,44,-41,20,-4).

Cf. A194647, A173783, A137432.

LinearRecurrence[{8, -26, 44, -41, 20, -4}, {64, 732, 4392, 18890, 66532, 205628}, 20] (* Jinyuan Wang, Feb 26 2020 *)

a(n) = -4*a(n-6) + 20*a(n-5) - 41*a(n-4) + 44*a(n-3) - 26*a(n-2) + 8*a(n-1).
G.f.: (1 + 56*x + 246*x^2 + 156*x^3 + 11*x^4)/((x-1)^4*(2*x-1)^2).
a(n) = (1771*n - 8709)*2^n + 235/3*n^3 + 880*n^2 + 12815/3*n + 8710.

A195594 Number of ways to place 6n nonattacking kings on a vertical cylinder 12 X 2n.

Original entry on oeis.org

128, 2102, 15988, 81606, 327192, 1118398, 3419648, 9643562, 25603228, 64923594, 158877948, 378088270, 879980720, 2011806182, 4532900488, 10091643138, 22244251284, 48622120786, 105526014500, 227633451206, 488451508168, 1043298475662, 2219419264848

Offset: 1

Vaclav Kotesovec, Sep 21 2011

nonn

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

Ray Chandler, Table of n, a(n) for n = 1..3296
V. Kotesovec, Non-attacking chess pieces
Index entries for linear recurrences with constant coefficients, signature (9, -34, 70, -85, 61, -24, 4).

Cf. A194648, A174154, A137432.

LinearRecurrence[{9,-34,70,-85,61,-24,4},{128,2102,15988,81606,327192,1118398,3419648},30] (* Harvey P. Dale, Aug 06 2024 *)

Recurrence: a(n) = 4*a(n-7) - 24*a(n-6) + 61*a(n-5) - 85*a(n-4) + 70*a(n-3) - 34*a(n-2) + 9*a(n-1).
G.f.: -(1 + 119*x + 984*x^2 + 1352*x^3 + 307*x^4 + 9*x^5)/((x-1)^5*(2*x-1)^2).
a(n) = (15839*n - 99729)*2^n + 231/2*n^4 + 1767*n^3 + 26001/2*n^2 + 53295*n + 99730.

A195595 Number of ways to place 7n nonattacking kings on a vertical cylinder 14 X 2n.

Original entry on oeis.org

256, 6060, 58776, 358564, 1649420, 6286658, 20984924, 63558566, 178909300, 476033636, 1212120160, 2980927200, 7129922604, 16675350430, 38293956836, 86629645122, 193553210580, 427974677968, 938053730248, 2040792091884, 4411561365324, 9483844861978

Offset: 1

Vaclav Kotesovec, Sep 21 2011

nonn

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

Ray Chandler, Table of n, a(n) for n = 1..3292
V. Kotesovec, Non-attacking chess pieces
Index entries for linear recurrences with constant coefficients, signature (10, -43, 104, -155, 146, -85, 28, -4).

Cf. A195004, A174155, A137432.

Recurrence: a(n) = -4*a(n-8) + 28*a(n-7) - 85*a(n-6) + 146*a(n-5) - 155*a(n-4) + 104*a(n-3) - 43*a(n-2) + 10*a(n-1).
G.f.: (1 + 246*x + 3543*x^2 + 9080*x^3 + 4915*x^4 + 442*x^5 + 15*x^6)/((x-1)^6*(2*x-1)^2).
a(n) = (157823*n - 1211433)*2^n + 9121/60*n^5 + 35581/12*n^4 + 352625/12*n^3 + 2179835/12*n^2 + 20456597/30*n + 1211434.

A195652 Number of ways to place 8n nonattacking kings on a 16 X 2n cylindrical chessboard.

Original entry on oeis.org

2304, 9476, 47058, 259372, 1536814, 9643562, 63558566, 437500380, 3130270224, 23174548666, 176740657340, 1382652697282, 11052082053262, 89954475408222, 743275585245898, 6219118726337532, 52583297643941856, 448492643088144992, 3853319870967662784

Offset: 1

Vaclav Kotesovec, Sep 22 2011

nonn

This cylinder is horizontal: a chessboard where it is supposed that rows 1 and 2n are in contact (number of columns = 16, number of rows = 2n).

Alex V. Breger, Table of n, a(n) for n = 1..1000
Artem M. Karavaev, Zealint blog (in Russian)
V. Kotesovec, Non-attacking chess pieces
Vaclav Kotesovec, Generating function
Index entries for linear recurrences with constant coefficients, order 208.

Cf. A137432, A174558, A195657.

Recurrence order is 208.

A195658 Number of ways to place 9n nonattacking kings on a vertical cylinder 18 X 2n.

Original entry on oeis.org

1024, 50922, 815816, 7238864, 44693472, 216134044, 877751236, 3130270224, 10105541204, 30179587994, 84719304384, 226268016376, 580363147336, 1440139184616, 3477556916828, 8210011147304, 19021962952188, 43385173057846, 97653259485592, 217359166880016

Offset: 1

Vaclav Kotesovec, Sep 22 2011

nonn

Vertical cylinder: a chessboard where it is supposed that the columns 1 and 18 are in contact (number of columns = 18, number of rows = 2n)

V. Kotesovec, Non-attacking chess pieces
Index entries for linear recurrences with constant coefficients, signature (12, -64, 200, -406, 560, -532, 344, -145, 36, -4).

Cf. A195653, A195648, A137432.

Recurrence: a(n) = -4*a(n-10) + 36*a(n-9) - 145*a(n-8) + 344*a(n-7) - 532*a(n-6) + 560*a(n-5) - 406*a(n-4) + 200*a(n-3) - 64*a(n-2) + 12*a(n-1).
G.f.: (1 + 1012*x + 38698*x^2 + 270088*x^3 + 503686*x^4 + 270112*x^5 + 37900*x^6 + 1516*x^7 + 25*x^8)/((x-1)^8*(2*x-1)^2).
a(n) = (21623809*n - 226349399)*2^n + 8913/40*n^7 + 124781/20*n^6 + 376359/4*n^5 + 977074*n^4 + 294753537/40*n^3 + 787733819/20*n^2 + 135269649*n + 226349400.

A195659 Number of ways to place 10n nonattacking kings on a vertical cylinder 20 X 2n.

Original entry on oeis.org

2048, 148352, 3076180, 33175486, 239238888, 1314160492, 5937279840, 23174548666, 80812754568, 257860425672, 766319864440, 2149806985106, 5753007728148, 14807729805472, 36902750545260, 89523360235366, 212335537312668, 494171055510052, 1131839140825580

Offset: 1

Vaclav Kotesovec, Sep 22 2011

nonn

Vertical cylinder: a chessboard where it is supposed that the columns 1 and 20 are in contact (number of columns = 20, number of rows = 2n).

V. Kotesovec, Non-attacking chess pieces
Index entries for linear recurrences with constant coefficients, signature (13, -76, 264, -606, 966, -1092, 876, -489, 181, -40, 4).

Cf. A195654, A195649, A137432.

Recurrence: a(n) = 4*a(n-11) - 40*a(n-10) + 181*a(n-9) - 489*a(n-8) + 876*a(n-7) - 1092*a(n-6) + 966*a(n-5) - 606*a(n-4) + 264*a(n-3) - 76*a(n-2) + 13*a(n-1).
G.f.: -(1 + 2035*x + 121804*x^2 + 1302988*x^3 + 3919832*x^4 + 3822444*x^5 + 1204400*x^6 + 113216*x^7 + 3167*x^8 + 13*x^9)/((x-1)^9*(2*x-1)^2).
a(n) = (296191755*n - 3519976573)*2^n + 524495/2016*n^8 + 4217363/504*n^7 + 2363921/16*n^6 + 66422455/36*n^5 + 557314865/32*n^4 + 8943856601/72*n^3 + 322704776641/504*n^2 + 90034143925/42*n + 3519976574.

A195660 Number of ways to place 11n nonattacking kings on a vertical cylinder 22 X 2n.

Original entry on oeis.org

4096, 433500, 11682296, 153802520, 1301236304, 8155899320, 41180193352, 176740657340, 668845118276, 2290966142762, 7241521734020, 21437333168798, 60123048359816, 161217291701134, 416373921218580, 1041997475699102, 2539265644237492, 6050425313244116

Offset: 1

Vaclav Kotesovec, Sep 22 2011

nonn

Vertical cylinder: a chessboard where it is supposed that the columns 1 and 22 are in contact (number of columns = 22, number of rows = 2n).

V. Kotesovec, Non-attacking chess pieces
Index entries for linear recurrences with constant coefficients, signature (14, -89, 340, -870, 1572, -2058, 1968, -1365, 670, -221, 44, -4).

Cf. A195655, A195650, A137432.

Recurrence: a(n) = -4*a(n-12) + 44*a(n-11) - 221*a(n-10) + 670*a(n-9) - 1365*a(n-8) + 1968*a(n-7) - 2058*a(n-6) + 1572*a(n-5) - 870*a(n-4) + 340*a(n-3) - 89*a(n-2) + 14*a(n-1).
G.f.: (1 + 4082*x + 376245*x^2 + 5977500*x^3 + 27440106*x^4 + 43897316*x^5 + 25742850*x^6 + 5340248*x^7 + 353057*x^8 + 5622*x^9 + 23*x^10)/((x-1)^10*(2*x-1)^2).
a(n) = (4480441703*n - 59644067185)*2^n + 10913705/36288*n^9 + 219791627/20160*n^8 + 6663742261/30240*n^7 + 1542837967/480*n^6 + 314791170001/8640*n^5 + 311982683023/960*n^4 + 6333872421866/2835*n^3 + 56561301500209/5040*n^2 + 46445710897861/1260*n + 59644067186.

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A195590 Number of ways to place 2n nonattacking kings on a vertical cylinder 4 X 2n.

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

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

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A195593 Number of ways to place 5n nonattacking kings on a vertical cylinder 10 X 2n.

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A195594 Number of ways to place 6n nonattacking kings on a vertical cylinder 12 X 2n.

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A195595 Number of ways to place 7n nonattacking kings on a vertical cylinder 14 X 2n.

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A195652 Number of ways to place 8n nonattacking kings on a 16 X 2n cylindrical chessboard.

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A195658 Number of ways to place 9n nonattacking kings on a vertical cylinder 18 X 2n.

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A195659 Number of ways to place 10n nonattacking kings on a vertical cylinder 20 X 2n.

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