A177758 - OEIS

%I #15 Aug 21 2024 10:51:45

%S 0,0,0,0,120,6912,52920,466944,1905120,8647680,25613280,81838080,

%T 198764280,510478080,1082161080,2393997312,4594961280,9120190464,

%U 16225246080,29656350720,49689816120,85128088320,135870624120

%N Number of ways to place 5 nonattacking bishops on an n X n toroidal board.

%H Vincenzo Librandi, <a href="/A177758/b177758.txt">Table of n, a(n) for n = 1..1000</a>

%H V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 6ed, 2013

%H <a href="/index/Rec#order_20">Index entries for linear recurrences with constant coefficients</a>, signature (2, 8, -18, -27, 72, 48, -168, -42, 252, 0, -252, 42, 168, -48, -72, 27, 18, -8, -2, 1).

%F 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).

%F 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).

%t 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 *)

%Y Cf. A172129, A177755, A177756, A177757.

%K nonn,easy

%O 1,5

%A _Vaclav Kotesovec_, May 13 2010