A322284 - OEIS

A322284

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

1, 4, 8, 22, 48, 116, 256, 584, 1280, 2832, 6144, 13344, 28672, 61504, 131072, 278656, 589824, 1245440, 2621440, 5505536, 11534336, 24118272, 50331648, 104859648, 218103808, 452988928, 939524096, 1946165248, 4026531840, 8321515520, 17179869184, 35433512960

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OFFSET

1,2

COMMENTS

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

Number of nonequivalent ways of placing n 2 X 2 tiles in an 3 X (2n+1) rectangle under all symmetry operations of the rectangle. - Andrew Howroyd, Dec 16 2018

Number of ways to choose modulo symmetry n vertices from a 1 X (2n-1) square grid with distances > sqrt(2) between the vertices. (Consider the interior 1 X (2*n-1) square grid of the 3 X (2n+1) square grid, or the square grid with the midpoints of the squares of the 2 X 2n chessboard as vertices.) - Wolfdieter Lang, Feb 07 2019

LINKS

Muniru A Asiru, Table of n, a(n) for n = 1..3280

Index entries for linear recurrences with constant coefficients, signature (4,-2,-8,8).

FORMULA

a(n) = (n+1)*2^(n-2) + (1 + (-1)^n)^(n/2 - 1) for n > 1.

a(n) = A238009(2*n+1, n). - Andrew Howroyd, Dec 16 2018

From Colin Barker, Dec 21 2018: (Start)

G.f.: x*(1 - 6*x^2 + 6*x^3) / ((1 - 2*x)^2*(1 - 2*x^2)).

a(n) = 4*a(n-1) - 2*a(n-2) - 8*a(n-3) + 8*a(n-4) for n>4. (End)

E.g.f.: (exp(2*x)*(1 + 2*x) + 2*cosh(sqrt(2)*x) - 3)/4. - Stefano Spezia, May 14 2023

EXAMPLE

For n = 2 there are a(2) = 4 distinct solutions from 12 that will not be repeated by all possible turns and reflections.