A364688 Number of 8-cycles in the hypercube graph Q_n.

0, 0, 0, 6, 696, 6720, 39840, 184800, 736512, 2644992, 8801280, 27624960, 82790400, 238977024, 668688384, 1822679040, 4858183680, 12700876800, 32647938048, 82682707968, 206650736640, 510425825280, 1247438438400, 3019527684096, 7245593051136, 17248655769600

Offset: 0

Author

Eric W. Weisstein, Aug 02 2023

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 (10,-40,80,-80,32).

Crossrefs

Cf. A001788 (4-cycles).

Cf. A290031 (6-cycles).

Programs

Mathematica

Table[Length[FindCycle[HypercubeGraph[n], {8}, All]], {n, 0, 9}]
Table[2^(n - 4) n (n - 1) (n - 2) (27 n - 79), {n, 0, 20}]
Table[3 2^(n - 3) Binomial[n, 3] (27 n - 79), {n, 0, 20}]
LinearRecurrence[{10, -40, 80, -80, 32}, {0, 0, 0, 6, 696}, 20]
CoefficientList[Series[6 x^3 (1 + 106 x)/(1 - 2 x)^5, {x, 0, 20}], x]

Formula

a(n) = 2^(n - 4)*n*(n - 1)*(n - 2)*(27*n - 79).

a(n) = 10*a(n-1) - 40*a(n-2) + 80*a(n-3) - 80*a(n-4) + 32*a(n-5).

G.f.: -6*x^3*(1 + 106*x)/(-1 + 2*x)^5.