A290779 - OEIS

%I #6 Aug 10 2017 12:58:06

%S 0,0,1,57,486,2360,8394,24354,61104,137412,283635,546403,994422,

%T 1725516,2875028,4625700,7219152,10969080,16276293,23645709,33705430,

%U 47228016,65154078,88618310,118978080,157844700,207117495,269020791,346143942,441484516,558494760

%N Number of 6-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_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).

%F a(n) = binomial(n + 1, 4)*(-62 + 11*n - 109*n^2 + 40*n^3)/70.

%F a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8).

%F G.f.: (x (x^2 + 49 x^3 + 58 x^4 + 12 x^5))/(-1 + x)^8.

%t Table[Binomial[n + 1, 4] (-62 + 11 n - 109 n^2 + 40 n^3)/70, {n, 20}]

%t LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {0, 0, 1, 57, 486, 2360, 8394, 24354}, 40]

%t CoefficientList[Series[(x^2 + 49 x^3 + 58 x^4 + 12 x^5)/(-1 + x)^8, {x, 0, 20}], x]

%o (PARI) a(n)=n*(40*n^6 - 189*n^5 + 189*n^4 + 105*n^3 - 105*n^2 + 84*n - 124)/1680 \\ _Charles R Greathouse IV_, Aug 10 2017

%Y Cf. A034827 (3-cycles), A051843 (4-cycles), A290775 (5-cycles).

%K nonn,easy

%O 1,4

%A _Eric W. Weisstein_, Aug 10 2017