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Number of 6-cycles in the n-hypercube graph.
2

%I #17 Aug 02 2023 17:17:26

%S 0,0,0,16,128,640,2560,8960,28672,86016,245760,675840,1802240,4685824,

%T 11927552,29818880,73400320,178257920,427819008,1016070144,2390753280,

%U 5578424320,12918456320,29712449536,67914170368,154350387200,348966092800,785173708800,1758789107712

%N Number of 6-cycles in the n-hypercube graph.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HypercubeGraph.html">Hypercube Graph</a>.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-24,32,-16).

%F a(n) = 2^(n + 1)*binomial(n, 3).

%F a(n) = 8*a(n-1)-24*a(n-2)+32*a(n-4)-16*a(n-4).

%F G.f.: (16*x^3)/(-1 + 2*x)^4.

%F From _Amiram Eldar_, Jan 05 2022: (Start)

%F Sum_{n>=3} 1/a(n) = 3*(2*log(2)-1)/16.

%F Sum_{n>=3} (-1)^(n+1)/a(n) = (3/2)^3*log(3/2) - 21/16. (End)

%t Table[2^(n + 1) Binomial[n, 3], {n, 0, 20}]

%t LinearRecurrence[{8, -24, 32, -16}, {0, 0, 0, 16}, 20]

%t CoefficientList[Series[(16 x^3)/(-1 + 2 x)^4, {x, 0, 20}], x]

%t Table[Length[FindCycle[HypercubeGraph[n], {6}, All]], {n, 0, 10}] (* _Eric W. Weisstein_, Aug 02 2023 *)

%o (Magma) [2^(n + 1)*Binomial(n, 3): n in [0..30]]; // _Wesley Ivan Hurt_, Apr 21 2021

%Y Cf. A001788 (4-cycles).

%Y Cf. A364688 (8-cycles).

%K nonn

%O 0,4

%A _Eric W. Weisstein_, Jul 17 2017