A177759 - OEIS

%I #19 Aug 20 2024 17:34:34

%S 0,0,0,0,0,2304,35280,811008,5080320,38784000,153679680,699678720,

%T 2120152320,7113012480,18036018000,49416536064,110279070720,

%U 261526745088,530024705280,1128038400000

%N Number of ways to place 6 nonattacking bishops on an n X n toroidal board.

%H Vincenzo Librandi, <a href="/A177759/b177759.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_24">Index entries for linear recurrences with constant coefficients</a>, signature (2, 10, -22, -44, 110, 110, -330, -165, 660, 132, -924, 0, 924, -132, -660, 165, 330, -110, -110, 44, 22, -10, -2, 1).

%F Explicit formula: 1/1440*(n-4)^2*(n-2)^2*n^2*(2*n^6 -36*n^5 +269*n^4 -1128*n^3 +3143*n^2-6330*n +7425 +(15*n^4 -240*n^3 +1545*n^2 -4950*n +6975)*(-1)^n).

%F G.f.: -48*x^6*(15*x^17 +2386*x^16 +6778*x^15 +133898*x^14 +235216*x^13 +1520054*x^12 +1844806*x^11 +5402462*x^10+4378450*x^9 +6819710*x^8 +3509350*x^7 +3079094*x^6+926032*x^5 +445642*x^4 +65754*x^3 +14946*x^2 +639*x+48)/((x-1)^13*(x+1)^11).

%t CoefficientList[Series[- 48 x^5 * (15 x^17 + 2386 x^16 + 6778 x^15 + 133898 x^14 + 235216 x^13 + 1520054 x^12 + 1844806 x^11 + 5402462 x^10 + 4378450 x^9 + 6819710 x^8 + 3509350 x^7 + 3079094 x^6 + 926032 x^5 + 445642 x^4 + 65754 x^3 + 14946 x^2 + 639 x + 48) / ((x - 1)^13  (x+1)^11), {x, 0, 50}], x] (* _Vincenzo Librandi_, May 31 2013 *)

%t LinearRecurrence[{2,10,-22,-44,110,110,-330,-165,660,132,-924,0,924,-132,-660,165,330,-110,-110,44,22,-10,-2,1},{0,0,0,0,0,2304,35280,811008,5080320,38784000,153679680,699678720,2120152320,7113012480,18036018000,49416536064,110279070720,261526745088,530024705280,1128038400000,2120098821120,4148067559680,7337013702480,13421018603520},30] (* _Harvey P. Dale_, Aug 20 2024 *)

%Y Cf. A176886, A177755, A177756, A177757, A177758.

%K nonn,easy

%O 1,6

%A _Vaclav Kotesovec_, May 13 2010