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

Author

Vaclav Kotesovec, Sep 21 2011

Keywords

• nonn
• easy
• changed

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

• 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).

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