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A172132 Number of ways to place 2 nonattacking knights on an n X n board.

%I A172132 #30 Dec 16 2024 14:27:45
%S A172132 0,6,28,96,252,550,1056,1848,3016,4662,6900,9856,13668,18486,24472,
%T A172132 31800,40656,51238,63756,78432,95500,115206,137808,163576,192792,
%U A172132 225750,262756,304128,350196,401302,457800,520056,588448,663366
%N A172132 Number of ways to place 2 nonattacking knights on an n X n board.
%D A172132 E. Bonsdorff, K. Fabel, O. Riihimaa, Schach und Zahl, 1966, p. 51-63
%H A172132 Vincenzo Librandi, <a href="/A172132/b172132.txt">Table of n, a(n) for n = 1..1000</a>
%H A172132 V. 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 A172132 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A172132 a(n) = (n - 1)*(n + 4)*(n^2 - 3*n + 4)/2.
%F A172132 G.f.: 2*(12*x^4-39*x^3+37*x^2-20*x+4)/(x-1)^5. - _Vaclav Kotesovec_, Mar 25 2010
%F A172132 a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - _Vincenzo Librandi_, Apr 30 2013
%F A172132 E.g.f.: (1/2)*(16 + (-16 + 16*x - 2*x^2 + 6*x^3 + x^4)*exp(x)). - _G. C. Greubel_, Apr 18 2022
%t A172132 Table[(n-1)(n+4)(n^2 -3n +4)/2, {n, 40}] (* _Vincenzo Librandi_, Apr 30 2013 *)
%o A172132 (Magma) I:=[0, 6, 28, 96, 252]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]]; // _Vincenzo Librandi_, Apr 30 2013
%o A172132 (Magma) [(n-1)*(n+4)*(n^2-3*n+4)/2: n in [1..40]]; // _Vincenzo Librandi_, Apr 30 2013
%o A172132 (SageMath) [(n-1)*(n+4)*(n^2-3*n+4)/2 for n in (1..40)] # _G. C. Greubel_, Apr 18 2022
%Y A172132 Cf. A036464, A172123.
%Y A172132 Column k=2 of A244081.
%K A172132 easy,nonn
%O A172132 1,2
%A A172132 _Vaclav Kotesovec_, Jan 26 2010