A290031 Number of 6-cycles in the n-hypercube graph.

0, 0, 0, 16, 128, 640, 2560, 8960, 28672, 86016, 245760, 675840, 1802240, 4685824, 11927552, 29818880, 73400320, 178257920, 427819008, 1016070144, 2390753280, 5578424320, 12918456320, 29712449536, 67914170368, 154350387200, 348966092800, 785173708800, 1758789107712

Offset: 0

Author

Eric W. Weisstein, Jul 17 2017

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Graph Cycle.
• Eric Weisstein's World of Mathematics, Hypercube Graph.
• Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Crossrefs

Cf. A001788 (4-cycles).

Cf. A364688 (8-cycles).

Programs

Magma

[2^(n + 1)*Binomial(n, 3): n in [0..30]]; // Wesley Ivan Hurt, Apr 21 2021

Mathematica

Table[2^(n + 1) Binomial[n, 3], {n, 0, 20}]
LinearRecurrence[{8, -24, 32, -16}, {0, 0, 0, 16}, 20]
CoefficientList[Series[(16 x^3)/(-1 + 2 x)^4, {x, 0, 20}], x]
Table[Length[FindCycle[HypercubeGraph[n], {6}, All]], {n, 0, 10}] (* Eric W. Weisstein, Aug 02 2023 *)

Formula

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

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

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

From Amiram Eldar, Jan 05 2022: (Start)

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

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