A348959 Childless terminal Wiener index of the rooted tree with Matula-Goebel number n.

0, 0, 0, 2, 0, 3, 2, 6, 4, 4, 0, 8, 3, 8, 5, 12, 2, 10, 6, 10, 10, 5, 4, 15, 6, 10, 12, 16, 4, 12, 0, 20, 6, 10, 12, 18, 8, 15, 12, 18, 3, 19, 8, 12, 14, 12, 5, 24, 20, 14, 12, 19, 12, 21, 7, 26, 18, 12, 2, 21, 10, 6, 22, 30, 14, 14, 6, 20, 14, 22, 10, 28, 10

Offset

1,4

Comments

This is a variation on the terminal Wiener index defined by Gutman, Furtula, and Petrović. Here terminal vertices are taken as the childless vertices, so a(n) is the sum of the path lengths between pairs of childless vertices.

This sequence differs from the free tree form A196055 when n is prime, since n prime means the root is degree 1 so is a terminal vertex for A196055 but not here.

Links

Kevin Ryde, Table of n, a(n) for n = 1..10000

F. Goebel, On a 1-1-Correspondence between Rooted Trees and Natural Numbers, Journal of Combinatorial Theory, series B, volume 29, 1980, pages 141-143.

Ivan Gutman, Boris Furtula and Miroslav Petrović, Terminal Wiener Index, Journal of Mathematical Chemistry, volume 46, 2009, pages 522-531.

D. W. Matula, A Natural Rooted Tree Enumeration By Prime Factorization, SIAM Review, volume 10, number 2, April 1968, page 273 (also at JSTOR).

Kevin Ryde, PARI/GP Code and Notes

Index entries for sequences related to Matula-Goebel numbers

Formula

a(n) = Sum_{j=1..k} a(primepi(p[j])) + E(p[j])*(C(n)-C(p[j]))), where n = p[1]*...*p[k] is the prime factorization of n with multiplicity (A027746), E(n) = A196048(n) is external path length, and C(n) = A109129(n) is number of childless vertices.

a(n) = A196055(n) - (A196048(n) if n prime).

a(n) = A196055(A288469(n)).

Prog

(PARI) See links.

Crossrefs

Cf. A196055 (free tree), A196048 (external path length), A109129 (childless vertices), A288469 (unplant).

Cf. A027746 (prime factorization).

Sequence in context: A266691 A083236 A345421 * A007492 A135351 A079451

Adjacent sequences: A348956 A348957 A348958 * A348960 A348961 A348962

Keyword

nonn

Author

Kevin Ryde, Nov 05 2021

Status

approved