A172129 Number of ways to place 5 nonattacking bishops on an n X n board.

0, 0, 0, 112, 3368, 39680, 282248, 1444928, 5865552, 20014112, 59673360, 159698416, 391202680, 890095584, 1902427800, 3853570560, 7450556064, 13829016768, 24759442464, 42930138864, 72328779720, 118747638592

Offset

1,4

Comments

For any fixed value of k>1, a(n) = n^(2k) /k! - 2n^(2k - 1) /3/(k - 2)! + ...

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, 2016-2020.

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,-10,-10,50,-34,-66,110,0,-110,66,34,-50,10,10,-6,1).

Formula

a(n) = n*(n-2)*(3*n^8 - 34*n^7 + 177*n^6 - 590*n^5 + 1435*n^4 - 2592*n^3 + 3326*n^2 - 2844*n + 1344)/360 if n is even.

a(n) = (n-1)*(n-2)*(n-3)*(3*n^7 - 22*n^6 + 80*n^5 - 204*n^4 + 379*n^3 - 464*n^2 + 378*n - 270)/360 if n is odd.

G.f.: 8*x^4*(14 + 337*x + 2574*x^2 + 9871*x^3 + 22040*x^4 + 31334*x^5 + 28808*x^6 + 17522*x^7 + 6666*x^8 + 1593*x^9 + 186*x^10 + 15*x^11) / ((1-x)^11*(1+x)^5). - Vaclav Kotesovec, Mar 25 2010

a(n) = (1/360)*(n-2)*( n*(1344 -2844*n +3326*n^2 -2592*n^3 +1435*n^4 -590*n^5 +177*n^6 -34*n^7 +3*n^8) -15*(54 -58*n +22*n^2 -3*n^3)*(1-(-1)^n)/2 ). - G. C. Greubel, Apr 17 2022

Mathematica

CoefficientList[Series[8*x^4*(14 +337*x +2574*x^2 +9871*x^3 +22040*x^4 +31334*x^5 +28808*x^6 +17522*x^7 +6666*x^8 +1593*x^9 +186*x^10 +15*x^11)/((1-x)^11*(1+x)^5), {x, 0, 50}], x]] (* Vincenzo Librandi, May 02 2013 *)

Prog

(SageMath) [(1/360)*(n-2)*( n*(1344 -2844*n +3326*n^2 -2592*n^3 +1435*n^4 -590*n^5 +177*n^6 -34*n^7 +3*n^8) -15*(54 -58*n +22*n^2 -3*n^3)*(n%2) ) for n in (1..50)] # G. C. Greubel, Apr 17 2022

Crossrefs

Cf. A108792, A172123, A172124, A172127.

Sequence in context: A267327 A008361 A271671 * A304552 A103860 A265660

Adjacent sequences: A172126 A172127 A172128 * A172130 A172131 A172132

Keyword

nonn,easy

Author

Vaclav Kotesovec, Jan 26 2010

Status

approved