A172124 Number of ways to place 3 nonattacking bishops on an n X n board.

0, 0, 26, 232, 1124, 3896, 10894, 26192, 56296, 110960, 204130, 355000, 589196, 940072, 1450134, 2172576, 3172944, 4530912, 6342186, 8720520, 11799860, 15736600, 20711966, 26934512, 34642744, 44107856, 55636594, 69574232

Offset: 1

Author

Vaclav Kotesovec, Jan 26 2010

Keywords

• nonn
• easy

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Christopher R. H. Hanusa, T. Zaslavsky, and S. Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853, a12016
• Vaclav Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
• Index entries for linear recurrences with constant coefficients, signature (6,-14,14,0,-14,14,-6,1).

Crossrefs

Cf. A047659, A172123.

Programs

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

Mathematica

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 *)

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

Formula

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

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

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)

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

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

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