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%I #34 Apr 29 2022 03:49:19
%S 0,0,0,0,4,14,36,76,140,234,364,536,756,1030,1364,1764,2236,2786,3420,
%T 4144,4964,5886,6916,8060,9324,10714,12236,13896,15700,17654,19764,
%U 22036,24476,27090,29884,32864,36036,39406,42980,46764,50764
%N Number of ways to place 3 nonattacking queens on a 3 X n board.
%H Vincenzo Librandi, <a href="/A061989/b061989.txt">Table of n, a(n) for n = 0..1000</a>
%H Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Ways of placing non-attacking queens and kings...</a>, part of "Between chessboard and computer", 1996, pp. 204 - 206.
%H E. Lucas, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k3943s/f256.image.r=">Recreations mathematiques I</a>, Albert Blanchard, Paris, 1992, p. 231.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F G.f.: 2*x^4*(2-x+2*x^2)/(1-x)^4.
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n >= 7.
%F Explicit formula (H. Tarry, 1890): a(n) = (n-3)*(n^2-6*n+12), n >= 3.
%F (4, 14, 36, ...) is the binomial transform of row 4 of A117937: (4, 10, 12, 6). - _Gary W. Adamson_, Apr 09 2006
%F a(n) = 2*A229183(n-3). - _R. J. Mathar_, Aug 16 2019
%F E.g.f.: 36 + 14*x + 2*x^2 + (-36 + 22*x - 6*x^2 + x^3)*exp(x). - _G. C. Greubel_, Apr 29 2022
%p A061989 := proc(n)
%p if n >= 3 then
%p (n-3)*(n^2-6*n+12) ;
%p else
%p 0;
%p end if;
%p end proc:
%p seq(A061989(n),n=0..30) ; # _R. J. Mathar_, Aug 16 2019
%t CoefficientList[Series[2*x^4*(2-x+2*x^2)/(1-x)^4, {x, 0, 50}], x] (* _Vincenzo Librandi_, May 02 2013 *)
%o (Magma) [0,0,0] cat [(n-3)*(n^2-6*n+12): n in [3..50]]; // _G. C. Greubel_, Apr 29 2022
%o (SageMath) [0,0,0]+[(n-3)*((n-3)^2 +3) for n in (3..50)] # _G. C. Greubel_, Apr 29 2022
%Y Cf. A061990, A117937, A229183.
%Y Essentially the same as A079908.
%K nonn,easy
%O 0,5
%A Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 29 2001
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