A292076 Number of 4-cycles in the n-Menger sponge graph.

0, 192, 5952, 137664, 2907456, 59398080, 1197999936, 24040445376, 481452900672, 9634211218368, 192725453785920, 3854838922404288, 77099417255585088, 1542009455673733056, 30840357998277021504, 616808511044877643200, 12336181029535005559104, 246723707059807997713344

Offset

1,2

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Eric Weisstein's World of Mathematics, Graph Cycle.

Eric Weisstein's World of Mathematics, Menger Sponge Graph.

Index entries for linear recurrences with constant coefficients, signature (31,-244,480).

Formula

From Andrew Howroyd, Jun 12 2025: (Start)

a(n) = Sum_{k=1..n-1} 8*3^(k-1)*A291066(n-k).

a(n) = 16*(5*20^n - 17*8^n + 12*3^n)/85. (End)

From Elmo R. Oliveira, Apr 10 2026: (Start)

a(n) = 31*a(n-1) - 244*a(n-2) + 480*a(n-3).

G.f.: 192*x^2/((1 - 20*x)*(1 - 8*x)*(1 - 3*x)).

E.g.f.: (16/85)*(12 - 17*exp(5*x) + 5*exp(17*x))*exp(3*x). (End)

Mathematica

A292076[n_] := 16*(5*20^n - 17*8^n + 12*3^n)/85; Array[A292076, 20] (* or *)
LinearRecurrence[{31, -244, 480}, {0, 192, 5952}, 20] (* Paolo Xausa, Feb 05 2026 *)

Prog

(PARI) a(n) = 16*(5*20^n - 17*8^n + 12*3^n)/85 \\ Andrew Howroyd, Jun 12 2025

Crossrefs

Cf. A291066 (edges), A292075 (6-cycles).

Sequence in context: A230751 A223375 A233154 * A223214 A215688 A173783

Adjacent sequences: A292073 A292074 A292075 * A292077 A292078 A292079

Keyword

nonn,easy

Author

Eric W. Weisstein, Sep 12 2017

Extensions

a(7) onwards from Andrew Howroyd, Jun 12 2025

Status

approved