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A172124
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Number of ways to place 3 nonattacking bishops on an n X n board.
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15
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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
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OFFSET
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1,3
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REFERENCES
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E. Bonsdorff, K. Fabel, O. Riihimaa, Schach und Zahl, 1966, p. 51-63
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LINKS
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(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
(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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