1,2

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

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).

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

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

1. 2. 3. 4.

----------------- ----------------- ----------------- -----------------

| * | | * | | | * | | | * | | * | | | | | * | | | |

----------------- ----------------- ----------------- -----------------

| | | | | | | | | | | | | * | | | | | | * |

----------------- ----------------- ----------------- -----------------

seq(coeff(series(x*(1-6*x^2+6*x^3)/((1-2*x)^2*(1-2*x^2)), x, n+1), x, n), n = 1 .. 35); # Muniru A Asiru, Dec 21 2018

(PARI) Vec(x*(1 - 6*x^2 + 6*x^3) / ((1 - 2*x)^2*(1 - 2*x^2)) + O(x^40)) \\ Colin Barker, Dec 21 2018

Cf. A001787, A061593, A061594, A238009, A321614.

Sequence in context: A153765 A003606 A048657 * A175655 A000639 A190795

Adjacent sequences: A322281 A322282 A322283 * A322285 A322286 A322287

nonn,easy

Anton Nikonov, Dec 02 2018

approved