A153642 - OEIS

A153642

a(n) = 4*n^2 + 24*n + 8.

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

1,1

COMMENTS

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

a(n) = A155135(2n+8) = A155136(2n+7).

a(n) = 4*A028881(n+3).

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

From Amiram Eldar, Mar 02 2023: (Start)

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)