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A172134 Number of ways to place 3 nonattacking knights on an n X n board.

%I A172134 #18 Apr 23 2022 18:45:45
%S A172134 0,4,36,276,1360,4752,13340,32084,68796,135040,247152,427380,705144,
%T A172134 1118416,1715220,2555252,3711620,5272704,7344136,10050900,13539552,
%U A172134 17980560,23570764,30535956,39133580,49655552,62431200,77830324
%N A172134 Number of ways to place 3 nonattacking knights on an n X n board.
%D A172134 E. Bonsdorff, K. Fabel, O. Riihimaa, Schach und Zahl, 1966, p. 51-63
%H A172134 Vincenzo Librandi, <a href="/A172134/b172134.txt">Table of n, a(n) for n = 1..1000</a>
%H A172134 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 A172134 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A172134 Explicit formula (Karl Fabel, 1966): a(n) = (n - 2)*(n + 5)*(n^4 - 3*n^3 - 8*n^2 + 66*n - 108)/6, for n >= 4.
%F A172134 G.f.: 4*x^2*(3*x^8-20*x^7+43*x^6-38*x^5+23*x^4-11*x^3-27*x^2-2*x-1)/(x-1)^7. - _Vaclav Kotesovec_, Mar 25 2010
%F A172134 From _G. C. Greubel_, Apr 18 2022: (Start)
%F A172134 a(n) = 7*a(n-1) -21*a(n-2) +35*a(n-3) -35*a(n-4) +21*a(n-5) -7*a(n-6) +a(n-7), for n >= 11.
%F A172134 E.g.f.: (1/6)*(-1080 - 312*x + 12*x^2 +13*x^3 + (1080 - 768*x + 228*x^2 + 38*x^4 + 15*x^5 + x^6)*exp(x)). (End)
%t A172134 CoefficientList[Series[4x(3x^8 -20x^7 +43x^6 -38x^5 +23x^4 -11x^3 -27x^2 -2x -1)/ (x-1)^7, {x, 0, 40}], x] (* _Vincenzo Librandi_, May 02 2013 *)
%o A172134 (Magma) [n le 3 select (n*(n-1))^2 else (n-2)*(n+5)*(n^4 -3*n^3 -8*n^2 +66*n -108)/6: n in [1..50]]; // _G. C. Greubel_, Apr 18 2022
%o A172134 (SageMath)
%o A172134 def A172134(n):
%o A172134     if (n<4): return (n*(n-1))^2
%o A172134     else: return (n-2)*(n+5)*(n^4 -3*n^3 -8*n^2 +66*n -108)/6
%o A172134 [A172134(n) for n in (1..50)] # _G. C. Greubel_, Apr 18 2022
%Y A172134 Cf. A047659, A172124, A172132.
%Y A172134 Column k=3 of A244081.
%K A172134 nonn,easy
%O A172134 1,2
%A A172134 _Vaclav Kotesovec_, Jan 26 2010