A368456 -id:A368456 - cp's OEIS frontend

A368190 Number of (undirected) cycles in the n-Dorogovtsev-Goltsev-Mendes graph.

0, 1, 11, 249, 74835, 5890739121, 34755832523270764251, 1207969003612007237832573159205646499369, 1459189113687796591938380205390010178829792070192521048490799792728844237848995

Offset: 0

Eric W. Weisstein, Dec 16 2023

nonn

Using the indexing convention that DGM(0) = P_2.

For n > 0, DGM(n) contains a unique longest cycle of length 3*2^(n-1).

Andrew Howroyd, Table of n, a(n) for n = 0..11
Eric Weisstein's World of Mathematics, Dorogovtsev-Goltsev-Mendes Graph.
Eric Weisstein's World of Mathematics, Graph Cycle.

Cf. A007018, A368456, A367967 (5-cycles), A367968 (6-cycles).

a(n) = {my(t=0,b=1); for(k=1, n, t = 3*t + b^3; b += b^2); t} \\ Andrew Howroyd, Dec 30 2023

a(n) = 3*a(n-1) + A007018(n-1)^3 for n > 0. - Andrew Howroyd, Dec 30 2023

Offset corrected and a(5) from Eric W. Weisstein, Dec 29 2023

Terms a(6) and beyond from Andrew Howroyd, Dec 30 2023

A387435 Number of dominating sets in the n-Dorogovtsev-Goltsev-Mendes graph.

Original entry on oeis.org

3, 7, 45, 13293, 461504710485, 37306936154345310416554765472710125

Offset: 0

Eric W. Weisstein, Aug 29 2025

a(6) has 104 decimal digits. - Andrew Howroyd, Aug 31 2025

Andrew Howroyd, Table of n, a(n) for n = 0..8
Eric Weisstein's World of Mathematics, Dominating Set.
Eric Weisstein's World of Mathematics, Dorogovtsev-Goltsev-Mendes Graph.

Cf. A115098 (domination number), A368456.

Join[{3}, Map[{1, 2, 1} . # &, NestList[Function[{p2, q1, q2}, {p2 (p2^2 + q1^2), q1^2 (q2 + p2), q2 (q1^2 + q2^2)}] @@ # &, {1, 2, 2}, 7]]] (* Eric W. Weisstein, Sep 03 2025 *)

step(v)={my([p2,q1,q2]=v); [p2*(p2^2+q1^2), q1^2*(q2+p2), q2*(q1^2+q2^2)]}
a(n)={if(n==0, 3, my(v=[1,2,2]); for(i=2, n, v=step(v)); v[1]+2*v[2]+v[3])} \\ Andrew Howroyd, Aug 31 2025

a(4) onwards from Andrew Howroyd, Aug 29 2025

cp's OEIS Frontend

A368190 Number of (undirected) cycles in the n-Dorogovtsev-Goltsev-Mendes graph.

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