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%I #38 Mar 26 2023 18:32:57
%S 12,79,408,1847,7698,30319,114606,419933,1501674,5266069,18174084,
%T 61892669,208424880,695179339,2299608732,7552444115,24648046806,
%U 79994460139,258339007890,830619734681,2660070154542,8488515938929,27000079296648,85629004867577
%N Number of ways to place 2n nonattacking kings on a 4 X 2n chessboard.
%H Bruno Berselli, <a href="/A061593/b061593.txt">Table of n, a(n) for n = 1..200</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">Non-attacking chess pieces</a>, 6ed, 2013, p. 90.
%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_04">Index entries for linear recurrences with constant coefficients</a>, signature (9,-28,33,-9).
%F G.f.: x*(12-29*x+33*x^2-9*x^3)/((1-3*x+x^2)*(1-3*x)^2).
%F a(n) = 9*a(n-1) - 28*a(n-2) + 33*a(n-3) - 9*a(n-4); a(1)=12, a(2)=79, a(3)=408, a(4)=1847.
%F a(n) = (17*n-109)*3^n + 2*Fibonacci(2*n+10).
%F a(n) = 17*A027471(n+2) - 126*A000244(n) + A025169(n+4).
%p with(combinat): A061593:=n->(17*n-109)*3^n+2*fibonacci(2*n+10): seq(A061593(n), n=1..30); # _Wesley Ivan Hurt_, Nov 08 2014
%t Table[(17 n - 109)*3^n + 2 Fibonacci[2 n + 10], {n, 30}] (* _Wesley Ivan Hurt_, Nov 08 2014 *)
%t CoefficientList[Series[x (12-29x+33x^2-9x^3)/((1-3x+x^2)(1-3x)^2),{x,0,30}],x] (* or *) LinearRecurrence[{9,-28,33,-9},{0,12,79,408,1847},30] (* _Harvey P. Dale_, Dec 20 2021 *)
%o (Magma) [(17*n-109)*3^n+2*Fibonacci(2*n+10): n in [1..30]]; // _Vincenzo Librandi_, Jul 12 2011
%Y Column k=2 of A350819.
%Y Cf. A061594, A001787.
%K nonn,easy,nice
%O 1,1
%A Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 22 2001
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