%I #44 May 14 2023 17:15:40

%S 1,4,8,22,48,116,256,584,1280,2832,6144,13344,28672,61504,131072,

%T 278656,589824,1245440,2621440,5505536,11534336,24118272,50331648,

%U 104859648,218103808,452988928,939524096,1946165248,4026531840,8321515520,17179869184,35433512960

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

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

%C 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

%C 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

%H Muniru A Asiru, <a href="/A322284/b322284.txt">Table of n, a(n) for n = 1..3280</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,-8,8).

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

%F a(n) = A238009(2*n+1, n). - _Andrew Howroyd_, Dec 16 2018

%F From _Colin Barker_, Dec 21 2018: (Start)

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

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

%F E.g.f.: (exp(2*x)*(1 + 2*x) + 2*cosh(sqrt(2)*x) - 3)/4. - _Stefano Spezia_, May 14 2023

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

%e 1. 2. 3. 4.

%e ----------------- ----------------- ----------------- -----------------

%e | * | | * | | | * | | | * | | * | | | | | * | | | |

%e ----------------- ----------------- ----------------- -----------------

%e | | | | | | | | | | | | | * | | | | | | * |

%e ----------------- ----------------- ----------------- -----------------

%p 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

%o (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

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

%K nonn,easy

%O 1,2

%A _Anton Nikonov_, Dec 02 2018