A061594 Number of ways to place 3n nonattacking kings on a 6 X 2n chessboard.

1, 32, 408, 3600, 26040, 166368, 976640, 5392704, 28432288, 144605184, 714611200, 3449705600, 16333065216, 76081271168, 349524164224, 1586790140800, 7130144209024, 31752978219904, 140298397039232, 615604372260736

Offset: 0

Author

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 22 2001

Keywords

• nonn
• nice

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• D. E. Knuth, Nonattacking kings on a chessboard, 1994.
• Vaclav Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes [_Vaclav Kotesovec_, Feb 06 2010]
• H. S. Wilf, The problem of the kings, Elec. J. Combin. 2, 1995.
• Index entries for linear recurrences with constant coefficients, signature (19, -148, 604, -1364, 1644, -928, 192).

Crossrefs

Column k=3 of A350819.

Cf. A001787, A061593.

Equals 231*A002697(n+1) - 2608*A000302(n) - 384*A000244(n) + 1103*A007070(n-1) + 780*A006012(n+1) + (n+1)*(17*A048580(n) + 12*A007070(n+1)).

Programs

PARI

a(n)=polcoeff((1+13*x-52*x^2-20*x^3+60*x^4-20*x^5)/((1-3*x)*(1-4*x)^2*(1-4*x+2*x^2)^2)+x*O(x^n),n)

Formula

G.f.: (1+13x-52x^2-20x^3+60x^4-20x^5)/((1-3x)(1-4x)^2(1-4x+2x^2)^2).

Explicit formula: (231n-2377)*4^n - 384*3^n + (1953*sqrt(2)/2+1381+(35*sqrt(2)+99/2)*n)*(2+sqrt(2))^n + (1381-1953*sqrt(2)/2+(99/2-35*sqrt(2))*n)*(2-sqrt(2))^n. - Vaclav Kotesovec, Feb 06 2010

Extensions

Corrected data by Vincenzo Librandi, Oct 12 2011