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A172200 Number of ways to place 2 nonattacking amazons (superqueens) on an n X n board.

%I A172200 #28 Jul 22 2025 07:32:48
%S A172200 0,0,0,20,92,260,580,1120,1960,3192,4920,7260,10340,14300,19292,25480,
%T A172200 33040,42160,53040,65892,80940,98420,118580,141680,167992,197800,
%U A172200 231400,269100,311220,358092,410060,467480,530720,600160,676192,759220,849660
%N A172200 Number of ways to place 2 nonattacking amazons (superqueens) on an n X n board.
%C A172200 A amazon (superqueen) moves like a queen and a knight.
%D A172200 Christian Poisson, Echecs et mathematiques, Rex Multiplex 29/1990, p.829
%H A172200 Vincenzo Librandi, <a href="/A172200/b172200.txt">Table of n, a(n) for n = 1..1000</a>
%H A172200 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 A172200 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A172200 Explicit formula (Christian Poisson, 1990): a(n) = (n - 1)(n - 2)(n - 3)(3n + 8)/6.
%F A172200 G.f.: 4*x^4*(5-2*x)/(1-x)^5. - _Colin Barker_, Jan 09 2013
%F A172200 E.g.f.: 8 + (1/6)*(-48 +48*x -24*x^2 +8*x^3 +3*x^4)*exp(x). - _G. C. Greubel_, Apr 28 2022
%t A172200 CoefficientList[Series[4x^3(5-2x)/(1-x)^5, {x, 0, 40}], x] (* _Vincenzo Librandi_, May 27 2013 *)
%t A172200 LinearRecurrence[{5,-10,10,-5,1},{0,0,0,20,92},40] (* or *) Table[(n-1)(n-2)(n-3)(3n+8)/6,{n,40}] (* _Harvey P. Dale_, May 16 2021 *)
%o A172200 (Magma) [(n-1)*(n-2)*(n-3)*(3*n+8)/6: n in [1..50]]; // _Vincenzo Librandi_, May 27 2013
%o A172200 (SageMath) [binomial(n-1,3)*(3*n+8) for n in (1..50)] # _G. C. Greubel_, Apr 28 2022
%Y A172200 Cf. A036464, A051223, A051224.
%K A172200 nonn,easy
%O A172200 1,4
%A A172200 _Vaclav Kotesovec_, Jan 29 2010