This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A348959 #15 Mar 22 2025 07:17:59
%S A348959 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,
%T A348959 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,
%U A348959 26,18,12,2,21,10,6,22,30,14,14,6,20,14,22,10,28,10
%N A348959 Childless terminal Wiener index of the rooted tree with Matula-Goebel number n.
%C A348959 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.
%C A348959 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.
%H A348959 Kevin Ryde, <a href="/A348959/b348959.txt">Table of n, a(n) for n = 1..10000</a>
%H A348959 F. Goebel, <a href="https://doi.org/10.1016/0095-8956(80)90049-0">On a 1-1-Correspondence between Rooted Trees and Natural Numbers</a>, Journal of Combinatorial Theory, series B, volume 29, 1980, pages 141-143.
%H A348959 Ivan Gutman, Boris Furtula and Miroslav Petrović, <a href="https://doi.org/10.1007/s10910-008-9476-2">Terminal Wiener Index</a>, Journal of Mathematical Chemistry, volume 46, 2009, pages 522-531.
%H A348959 D. W. Matula, <a href="https://doi.org/10.1137/1010054">A Natural Rooted Tree Enumeration By Prime Factorization</a>, SIAM Review, volume 10, number 2, April 1968, page 273 (also <a href="http://www.jstor.org/stable/2027327">at JSTOR</a>).
%H A348959 Kevin Ryde, <a href="/A348959/a348959.gp.txt">PARI/GP Code and Notes</a>
%H A348959 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A348959 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.
%F A348959 a(n) = A196055(n) - (A196048(n) if n prime).
%F A348959 a(n) = A196055(A288469(n)).
%o A348959 (PARI) \\ See links.
%Y A348959 Cf. A196055 (free tree), A196048 (external path length), A109129 (childless vertices), A288469 (unplant).
%Y A348959 Cf. A027746 (prime factorization).
%K A348959 nonn
%O A348959 1,4
%A A348959 _Kevin Ryde_, Nov 05 2021