|
%I #19 Jul 14 2017 13:25:22
%S 0,0,15,96,330,855,1866,3624,6468,10818,17193,26208,38598,55209,77028,
%T 105168,140904,185652,241011,308736,390786,489291,606606,745272,
%U 908076,1098006,1318317,1572480,1864254,2197629,2576904,3006624,3491664,4037160,4648599,5331744,6092730
%N Number of 4-cycles in the n-triangular honeycomb queen graph.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (4, -4, -4, 10, -4, -4, 4, -1).
%F a(n) = (24*n^5 + 170*n^4 - 660*n^3 + 160*n^2 + 606*n - 165 + (-1)^n*(165 - 30*n))/320.
%F a(n) = 4*a(n-1)-4*a(n-2)-4*a(n-3)+10*a(n-4)-4*a(n-5)-4*a(n-6)+4*a(n-7)-a(n-8).
%F G.f.: (-3*x^3*(-5 - 12*x - 2*x^2 + 7*x^3))/((-1 + x)^6*(1 + x)^2).
%t Table[(24 n^5 + 170 n^4 - 660 n^3 + 160 n^2 + 606 n - 165 + (-1)^n (165 - 30 n))/320, {n, 20}]
%t LinearRecurrence[{4, -4, -4, 10, -4, -4, 4, -1}, {0, 0, 15, 96, 330, 855, 1866, 3624}, 20]
%t CoefficientList[Series[-((3 x^2 (-5 - 12 x - 2 x^2 + 7 x^3))/((-1 + x)^6 (1 + x)^2)), {x, 0, 20}], x]
%Y Cf. A105636 (3-cycles), A289706 (5-cycles), A289707 (6-cycles).
%K nonn
%O 1,3
%A _Eric W. Weisstein_, Jul 14 2017
|
