%I #27 Aug 07 2024 09:20:33

%S 1,32,408,3600,26040,166368,976640,5392704,28432288,144605184,

%T 714611200,3449705600,16333065216,76081271168,349524164224,

%U 1586790140800,7130144209024,31752978219904,140298397039232,615604372260736

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

%H Vincenzo Librandi, <a href="/A061594/b061594.txt">Table of n, a(n) for n = 0..1000</a>

%H D. E. Knuth, <a href="http://www-cs-faculty.stanford.edu/~knuth/preprints.html">Nonattacking kings on a chessboard</a>, 1994.

%H Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Number of ways of placing non-attacking queens and kings on boards of various sizes</a> [_Vaclav Kotesovec_, Feb 06 2010]

%H H. S. Wilf, <a href="https://doi.org/10.37236/1197">The problem of the kings</a>, Elec. J. Combin. 2, 1995.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (19, -148, 604, -1364, 1644, -928, 192).

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

%F 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

%o (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)

%Y Column k=3 of A350819.

%Y Cf. A001787, A061593.

%Y 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)).

%K nonn,nice

%O 0,2

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

%E Corrected data by _Vincenzo Librandi_, Oct 12 2011