A307457 -id:A307457 - cp's OEIS frontend

A307549 Number of longest paths in the n-Apollonian network.

12, 174, 3084, 35971776, 2752830231727104, 9068533325994694179911466663936, 55357290942669201825421134704769198540735968154850291354435584

Offset: 1

Eric W. Weisstein, Apr 14 2019

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..9
Eric Weisstein's World of Mathematics, Apollonian Network.
Eric Weisstein's World of Mathematics, Longest Path.

Cf. A292002, A307457 (longest path lengths).

a(5) onwards from Andrew Howroyd, Jun 09 2025

A309524 Triangle read by rows: T(n,k) is the number of simple connected graphs on n nodes with longest path having k nodes, (1 <= k <= n).

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 5, 0, 0, 1, 2, 18, 0, 0, 1, 3, 17, 91, 0, 0, 1, 3, 29, 86, 734, 0, 0, 1, 4, 42, 176, 864, 10030, 0, 0, 1, 4, 64, 309, 2032, 10243, 248427

Offset: 1

Andrew Howroyd, Sep 06 2019

Paths here are subgraphs that are isomorphic to a path graph and are measured by the number of vertices they contain rather than the number of edges. No vertex can appear more than once.

Paths with three vertices exist in all connected graphs with at least three vertices. For n > 3, the star graph is the only graph in which longer paths are not possible.

			Triangle begins:
  1;
  0, 1;
  0, 0, 2;
  0, 0, 1, 5;
  0, 0, 1, 2, 18;
  0, 0, 1, 3, 17,  91;
  0, 0, 1, 3, 29,  86,  734;
  0, 0, 1, 4, 42, 176,  864, 10030;
  0, 0, 1, 4, 64, 309, 2032, 10243, 248427;
  ...

Eric Weisstein's World of Mathematics, Longest Path
Eric Weisstein's World of Mathematics, Traceable Graph

Row sums are A001349.
Right diagonal is A057864.
Cf. A325455 (circumference = longest cycle).
Cf. A307457.

cp's OEIS Frontend

A307549 Number of longest paths in the n-Apollonian network.

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