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A035286
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Number of ways to place a non-attacking white and black king on n X n chessboard.
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3
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0, 0, 32, 156, 456, 1040, 2040, 3612, 5936, 9216, 13680, 19580, 27192, 36816, 48776, 63420, 81120, 102272, 127296, 156636, 190760, 230160, 275352, 326876, 385296, 451200, 525200, 607932, 700056, 802256, 915240, 1039740, 1176512, 1326336
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OFFSET
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1,3
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COMMENTS
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A legal position is such that the kings are not on (horizontal, vertical or diagonal) neighboring squares.
For n < 3 this is not possible, for n >= 3 a king on the corner, border or elsewhere on the board takes away 4, 6 resp. 9 allowed squares from the n X n board, which yields the formula. - M. F. Hasler, Nov 17 2021
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LINKS
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FORMULA
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a(n) = n^4 - 9 n^2 + 12 n - 4.
G.f.: x^3*(8 - x - x^2)/(1 - x)^5. - Colin Barker, Jan 09 2013
a(n) = (n - 1) (n - 2) (n^2 + 3 n - 2). - M. F. Hasler, Nov 17 2021
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) - Nathan L. Skirrow, Oct 11 2022
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EXAMPLE
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There are 32 ways of putting 2 distinct kings on a 3 X 3 board so that neither can capture the other.
The first nonzero term occurs for n = 3 where we have the possibilities
K x O x K x
x x O and x x x and rotations of these by +-90 degrees and 180 degrees,
O O O O O O
where 'x' are forbidden squares, and 'O' are squares the opposite king can be placed on. This yields the a(3) = 4*(5 + 3) = 32 possibilities. (End)
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MATHEMATICA
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CoefficientList[Series[4 x^2 (x^2 + x - 8)/(x - 1)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Oct 20 2013 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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