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%I #19 Jul 27 2017 09:08:19
%S 0,0,16,911,8013,38130,129932,358272,851710,1815124,3554910,6510729,
%T 11289019,18704640,29823436,46014402,69002190,100930284,144424446,
%U 202667301,279473821,379377584,507719550,670746120,875712560,1130992902,1446199474,1832304547,2301777585,2868718404
%N Number of 6-cycles in the n-triangular honeycomb queen graph.
%H Colin Barker, <a href="/A289707/b289707.txt">Table of n, a(n) for n = 1..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>
%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (3, 1, -10, 3, 13, -3, -12, -3, 13, 3, -10, 1, 3, -1).
%F a(n) = 3*a(n-1)+a(n-2)-10*a(n-3)+3*a(n-4)+13*a(n-5)-3*a(n-6)-12*a(n-7)-3*a(n-8)+13*a(n-9)+3*a(n-10)-10*a(n-11)+a(n-12)+3*a(n-13)-a(n-14).
%F G.f.: x^3*(16 + 863*x + 5264*x^2 + 13340*x^3 + 16591*x^4 + 7535*x^5 - 7572*x^6 - 14592*x^7 - 9919*x^8 - 2886*x^9) / ((1 - x)^8*(1 + x)^4*(1 + x + x^2)). - _Colin Barker_, Jul 27 2017
%t Table[(-315 (-1)^n (-2489 + 1659 n - 297 n^2 + 10 n^3) - 792995 + 3789081 n - 1968939 n^2 - 3033450 n^3 + 3489990 n^4 - 1269366 n^5 + 154014 n^6 + 1440 n^7 + 8960 Cos[2 n Pi/3] - 8960 Sqrt[3] Sin[2 n Pi/3])/40320, {n, 20}]
%t LinearRecurrence[{3, 1, -10, 3, 13, -3, -12, -3, 13, 3, -10, 1, 3, -1}, {0, 0, 16, 911, 8013, 38130, 129932, 358272, 851710, 1815124, 3554910, 6510729, 11289019, 18704640}, 20]
%o (PARI) concat(vector(2), Vec(x^3*(16 + 863*x + 5264*x^2 + 13340*x^3 + 16591*x^4 + 7535*x^5 - 7572*x^6 - 14592*x^7 - 9919*x^8 - 2886*x^9) / ((1 - x)^8*(1 + x)^4*(1 + x + x^2)) + O(x^60))) \\ _Colin Barker_, Jul 27 2017
%Y Cf. A105636 (3-cycles), A289705 (4-cycles), A289706 (5-cycles).
%K nonn,easy
%O 1,3
%A _Eric W. Weisstein_, Jul 14 2017
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