A177758 Number of ways to place 5 nonattacking bishops on an n X n toroidal board.
0, 0, 0, 0, 120, 6912, 52920, 466944, 1905120, 8647680, 25613280, 81838080, 198764280, 510478080, 1082161080, 2393997312, 4594961280, 9120190464, 16225246080, 29656350720, 49689816120, 85128088320, 135870624120
Offset: 1
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- V. Kotesovec, Non-attacking chess pieces, 6ed, 2013
- Index entries for linear recurrences with constant coefficients, signature (2, 8, -18, -27, 72, 48, -168, -42, 252, 0, -252, 42, 168, -48, -72, 27, 18, -8, -2, 1).
Programs
-
Mathematica
CoefficientList[Series[- 24 x^4 (5 x^14 + 406 x^13 + 1333 x^12 + 14880 x^11 + 24307 x^10 + 97498 x^9 + 95187 x^8 + 175328 x^7 + 100307 x^6 + 93018 x^5 + 28147 x^4 + 12832 x^3 + 1589 x^2 + 278 x + 5) / ((x - 1)^11 (x + 1)^9), {x, 0, 50}], x] (* Vincenzo Librandi, May 31 2013 *)
Formula
Explicit formula: 1/240*(n-4)^2*(n-2)^2*n^2*(2n^4 -16n^3 +54n^2 -108n+153 +(10n^2 -60n +135)*(-1)^n).
G.f.: -24x^5*(5x^14 +406x^13 +1333x^12 +14880x^11 +24307x^10 +97498x^9 +95187x^8 +175328x^7 +100307x^6 +93018x^5 +28147x^4 +12832x^3 +1589x^2 +278x+5)/((x-1)^11*(x+1)^9).