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A322284 Number of nonequivalent ways to place n nonattacking kings on a 2 X 2n chessboard under all symmetry operations of the rectangle. 3
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 (list; graph; refs; listen; history; text; internal format)
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)

EXAMPLE

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.

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

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

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

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

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

MAPLE

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

PROG

(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

CROSSREFS

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

Sequence in context: A153765 A003606 A048657 * A175655 A000639 A190795

Adjacent sequences:  A322281 A322282 A322283 * A322285 A322286 A322287

KEYWORD

nonn,easy

AUTHOR

Anton Nikonov, Dec 02 2018

STATUS

approved

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Last modified January 23 19:12 EST 2020. Contains 331175 sequences. (Running on oeis4.)