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A348959
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Childless terminal Wiener index of the rooted tree with Matula-Goebel number n.
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4
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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
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OFFSET
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1,4
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COMMENTS
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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.
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LINKS
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Ivan Gutman, Boris Furtula and Miroslav Petrović, Terminal Wiener Index, Journal of Mathematical Chemistry, volume 46, 2009, pages 522-531.
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FORMULA
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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.
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PROG
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(PARI) See links.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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