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A153642
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a(n) = 4*n^2 + 24*n + 8.
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5
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36, 72, 116, 168, 228, 296, 372, 456, 548, 648, 756, 872, 996, 1128, 1268, 1416, 1572, 1736, 1908, 2088, 2276, 2472, 2676, 2888, 3108, 3336, 3572, 3816, 4068, 4328, 4596, 4872, 5156, 5448, 5748, 6056, 6372, 6696, 7028, 7368, 7716, 8072, 8436, 8808, 9188
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OFFSET
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1,1
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COMMENTS
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2*(fifth subdiagonal of triangle A144562).
Sequence gives values x of solutions (x, y) to the Diophantine equation x^3+28*x^2 = y^2. For a more comprehensive list of solutions see A155135.
For n >= 3, a(n - 1) is the number of checkmate positions with white queen and white king against black king on an n X n board. Reason: The black king can only be on the edge. There are 4*(4*n + 1) checkmate positions where the black king is in the corner, 4*(2*n + 4) checkmate positions where the black king is immediately adjacent to the corner square, and there are 4*(n - 4)*(n + 2) checkmate positions where the black king is on another edge square. That's a total of 4*n^2 + 16*n - 12 = a(n - 1) checkmate positions. - Felix Huber, Oct 29 2023
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LINKS
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FORMULA
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G.f.: 4*(3 - x)*(3 - 2*x)/(1-x)^3.
a(n)= 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: 4*(-2 + (2 + 7*x + x^2)*exp(x)). - G. C. Greubel, Aug 23 2016
Sum_{n>=1} 1/a(n) = 1/56 - cot(sqrt(7)*Pi)*Pi/(8*sqrt(7)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 31/168 - cosec(sqrt(7)*Pi)*Pi/(8*sqrt(7)). (End)
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MATHEMATICA
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PROG
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(Magma) [ 4*(n+3)^2-28: n in [1..45] ];
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CROSSREFS
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KEYWORD
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nonn,easy,less
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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