A061594 -id:A061594 - cp's OEIS frontend

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

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

A321614 Number of nonequivalent ways to place 2n nonattacking kings on a 4 X 2n chessboard under all symmetry operations of the rectangle.

Original entry on oeis.org

1, 4, 23, 106, 473, 1939, 7618, 28703, 105112, 375597, 1316944, 4544124, 15474559, 52108212, 173799309, 574908646, 1888125243, 6162032375, 19998659760, 64584817367, 207655073310, 665017743665

Offset: 0

Anton Nikonov, Dec 19 2018

A maximum of 2n nonattacking kings can be placed on a 4 X 2n chessboard.

Number of nonequivalent ways of placing 2n 2 X 2 tiles in an 5 X (2n+1) rectangle under all symmetry operations of the rectangle. - Andrew Howroyd, Dec 21 2018

Cf. A001787, A061593, A061594, A231145, A319096, A322284.

a(n) = A231145(2*n+1, 2n).
Conjectures from Colin Barker, Dec 22 2018: (Start)
G.f.: (1 - 2*x)*(1 - 6*x + 17*x^2 - 18*x^3 - 2*x^4 + 7*x^5 + 6*x^6 - 3*x^7) / ((1 - x)^2*(1 - 3*x)^2*(1 - 3*x + x^2)*(1 - x - x^2)*(1 - 3*x^2)).
a(n) = 12*a(n-1) - 54*a(n-2) + 98*a(n-3) + 17*a(n-4) - 346*a(n-5) + 505*a(n-6) - 210*a(n-7) - 120*a(n-8) + 126*a(n-9) - 27*a(n-10) for n>9.
(End)

cp's OEIS Frontend

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

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