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A172201 Number of ways to place 3 nonattacking amazons (superqueens) on an n X n board.

%I A172201 #21 Apr 29 2022 03:48:47
%S A172201 0,0,0,0,48,424,1976,6616,17852,41544,86660,166288,298616,508200,
%T A172201 827168,1296744,1968676,2907016,4189772,5910944,8182400,11136168,
%U A172201 14926536,19732600,25760588,33246664,42459476,53703216,67320392,83695144
%N A172201 Number of ways to place 3 nonattacking amazons (superqueens) on an n X n board.
%C A172201 An amazon (superqueen) moves like a queen and a knight.
%D A172201 Panos Louridas, idee & form 93/2007, pp. 2936-2938.
%H A172201 Vincenzo Librandi, <a href="/A172201/b172201.txt">Table of n, a(n) for n = 1..1000</a>
%H A172201 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>
%H A172201 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (5,-8,0,14,-14,0,8,-5,1).
%F A172201 Explicit formula (Panos Louridas, 2007): a(n) = (2*n^6 - 20*n^5 + 31*n^4 + 314*n^3 - 1452*n^2 + 2040*n - 672)/12 if n is even (n >= 4) and a(n) = (2*n^6 - 20*n^5 + 31*n^4 + 314*n^3 - 1452*n^2 + 2034*n - 669)/12 if n is odd (n >= 5).
%F A172201 G.f.: 4*x^5*(12+46*x+60*x^2+32*x^3-23*x^4-13*x^5+7*x^6-x^7)/((1+x)^2*(1-x)^7). - _Vaclav Kotesovec_, Mar 24 2010
%F A172201 a(n) = (1/24)*(4*n^6 - 40*n^5 + 62*n^4 + 628*n^3 - 2904*n^2 + 4074*n - 1341 + 3*(-1)^n*(2*n-1)) - 20*[n=1] - 8*[n=2] + 4*[n=3]. - _G. C. Greubel_, Apr 29 2022
%t A172201 CoefficientList[Series[4*x^5*(12+46*x+60*x^2+32*x^3-23*x^4-13*x^5+7*x^6-x^7)/((1+x)^2*(1-x)^7), {x, 0, 40}], x] (* _Vincenzo Librandi_, May 27 2013 *)
%o A172201 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0,0,0,0] cat Coefficients(R!( 4*x^5*(12+46*x+60*x^2+32*x^3-23*x^4-13*x^5+7*x^6-x^7)/((1+x)^2*(1-x)^7 ))); // _G. C. Greubel_, Apr 29 2022
%o A172201 (SageMath) [(1/24)*(4*n^6 -40*n^5 +62*n^4 +628*n^3 -2904*n^2 +4074*n -1341 +3*(-1)^n*(2*n-1)) -4*(5*bool(n==1) +2*bool(n==2) -bool(n==3)) for n in (1..40)] # _G. C. Greubel_, Apr 29 2022
%Y A172201 Cf. A047659, A051223, A051224, A061989, A172200.
%K A172201 nonn,easy
%O A172201 1,5
%A A172201 _Vaclav Kotesovec_, Jan 29 2010