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%I #8 Aug 10 2017 12:55:25
%S 0,0,2,24,138,532,1596,4032,8988,18216,34254,60632,102102,164892,
%T 256984,388416,571608,821712,1156986,1599192,2174018,2911524,3846612,
%U 5019520,6476340,8269560,10458630,13110552,16300494,20112428,24639792,29986176,36266032,43605408,52142706
%N Number of 5-cycles in the n-triangular honeycomb bishop 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_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F a(n) = 2/5 * binomial(n + 1, 4)*(8 - 7*n + 2*n^2).
%F a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
%F G.f.: -((2 x (x^2 + 5 x^3 + 6 x^4))/(-1 + x)^7).
%t Table[2/5 Binomial[n + 1, 4] (8 - 7 n + 2 n^2), {n, 20}]
%t LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 2, 24, 138, 532, 1596}, 20]
%t CoefficientList[Series[-((2 (x^2 + 5 x^3 + 6 x^4))/(-1 + x)^7), {x, 0, 20}], x]
%o (PARI) a(n)=n*(2*n^5 - 11*n^4 + 20*n^3 - 5*n^2 - 22*n + 16)/60 \\ _Charles R Greathouse IV_, Aug 10 2017
%Y Cf. A034827 (3-cycles in the triangular honeycomb bishop graph), A051843 (4-cycles), A290779 (6-cycles).
%K nonn,easy
%O 1,3
%A _Eric W. Weisstein_, Aug 10 2017
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