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A196055
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The terminal Wiener index of the rooted tree with Matula-Goebel number n.
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2
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0, 1, 2, 2, 3, 3, 6, 6, 4, 4, 4, 8, 8, 8, 5, 12, 8, 10, 12, 10, 10, 5, 10, 15, 6, 10, 12, 16, 10, 12, 5, 20, 6, 10, 12, 18, 15, 15, 12, 18, 10, 19, 16, 12, 14, 12, 12, 24, 20, 14, 12, 19, 20, 21, 7, 26, 18, 12, 10, 21, 18, 6, 22, 30, 14, 14, 15, 20, 14, 22, 18, 28, 19, 18, 16, 26, 14, 22, 12, 28, 24, 12, 12, 30, 14, 19, 14, 21, 24, 24
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OFFSET
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1,3
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COMMENTS
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The terminal Wiener index of a connected graph is the sum of the distances between all pairs of nodes of degree 1.
The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m >= 2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.
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LINKS
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FORMULA
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Let LV(m) and EPL(m) denote the number of leaves and the external path length, respectively, of the rooted tree with Matula number m (see A109129 and A196048, where LV(m) and EPL(m) are obtained recursively). a(1)=0; if n=p(t) (=the t-th prime) and t is prime, then a(n) = a(t) + LV(t); if n=p(t) (=the t-th prime) and t is not prime, then a(n) = a(t) + LV(t) + EPL(t). Now assume that n is not prime; it can be written n=rs, where r is prime and s >= 2. If s is prime, then a(n) = a(r) - EPL(r) + a(s) - EPL(s) + EPL(r)*LV(s) + EPL(s)*LV(r); if s is not prime, then a(n) = a(r) - EPL(r) + a(s) + EPL(r)*LV(s) + EPL(s)*LV(r); the Maple program is based on this recursive formula.
If m > 2 then a(2^m) = m(m-1) because the rooted tree with Matula-Goebel number 2^m is a star with m edges and the vertices of each of the binomial(m,2) pairs of nodes of degree 1 are at distance 2.
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EXAMPLE
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a(7)=6 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (2+2+2=6).
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MAPLE
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with(numtheory): TW := proc (n) local r, s, LV, EPL, Tw: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: LV := proc (n) if n = 1 then 0 elif n = 2 then 1 elif bigomega(n) = 1 then LV(pi(n)) else LV(r(n))+LV(s(n)) end if end proc: EPL := proc (n) if n = 1 then 0 elif n = 2 then 1 elif bigomega(n) = 1 then EPL(pi(n))+LV(pi(n)) else EPL(r(n))+EPL(s(n)) end if end proc: Tw := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then Tw(pi(n)) else Tw(r(n))+Tw(s(n))+EPL(r(n))*LV(s(n))+EPL(s(n))*LV(r(n)) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 and bigomega(pi(n)) = 1 then TW(pi(n))+LV(pi(n)) elif bigomega(n) = 1 then TW(pi(n))+EPL(n) else Tw(r(n))+Tw(s(n))+EPL(r(n))*LV(s(n))+EPL(s(n))*LV(r(n)) end if end proc; seq(TW(n), n = 1 .. 90);
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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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