A172124 - OEIS

%I #18 Apr 17 2022 09:23:06

%S 0,0,26,232,1124,3896,10894,26192,56296,110960,204130,355000,589196,

%T 940072,1450134,2172576,3172944,4530912,6342186,8720520,11799860,

%U 15736600,20711966,26934512,34642744,44107856,55636594,69574232

%N Number of ways to place 3 nonattacking bishops on an n X n board.

%D E. Bonsdorff, K. Fabel, O. Riihimaa, Schach und Zahl, 1966, p. 51-63

%H Vincenzo Librandi, <a href="/A172124/b172124.txt">Table of n, a(n) for n = 1..1000</a>

%H Christopher R. H. Hanusa, T. Zaslavsky, and S. Chaiken, <a href="http://arxiv.org/abs/1609.00853">A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks)</a>, arXiv preprint arXiv:1609.00853, a12016

%H 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 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (6,-14,14,0,-14,14,-6,1).

%F Explicit formulas (Karl Fabel, 1966): (Start)

%F a(n) = n*(n-2)*(2*n^4 - 4*n^3 + 7*n^2 - 6*n + 4)/12 if n is even.

%F a(n) = (n-1)*(2*n^5 - 6*n^4 + 9*n^3 - 11*n^2 + 5*n - 3)/12 if n is odd. (End)

%F G.f.: 2*x^3*(13+38*x+48*x^2+18*x^3+3*x^4)/((1-x)^7*(1+x)). - ._Vaclav Kotesovec_, Mar 25 2010

%F a(n) = (2*(n-2)*n*(2*n^4-4*n^3+7*n^2-6*n+4)-3*(-1)^n+3)/24. - _Bruno Berselli_, May 26 2013

%F E.g.f.: (1/24)*( (3 - 6*x + 6*x^2 + 100*x^3 + 130*x^4 + 44*x^5 + 4*x^6)*exp(x) - 3*exp(-x) ). - _G. C. Greubel_, Apr 16 2022

%t CoefficientList[Series[2x^2(3x^4 +18x^3 +48x^2 +38x +13)/((1-x)^7 (x+1)), {x, 0, 30}], x] (* _Vincenzo Librandi_, May 26 2013 *)

%o (Magma) [(n*(n-2)*(2*n^4 -4*n^3 +7*n^2 -6*n +4) +3*(n mod 2))/12: n in [1..40]]; // _G. C. Greubel_, Apr 16 2022

%o (SageMath) [(n*(n-2)*(2*n^4 -4*n^3 +7*n^2 -6*n +4) +3*(n%2))/12 for n in (1..40)] # _G. C. Greubel_, Apr 16 2022

%Y Cf. A047659, A172123.

%K nonn,easy

%O 1,3

%A _Vaclav Kotesovec_, Jan 26 2010