

A196055


The terminal Wiener index of the rooted tree with MatulaGoebel number n.


0



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

1,3


COMMENTS

The terminal Wiener index of a connected graph is the sum of the distances between all pairs of nodes of degree 1.
The MatulaGoebel number of a rooted tree can be defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the MatulaGoebel 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 MatulaGoebel numbers of the m branches of T.


REFERENCES

F. Goebel, On a 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman, B. Furtula and M. Petrovic, Terminal Wiener index, J. Math. Chem., 46. 2009, 522531.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.


LINKS

Table of n, a(n) for n=1..90.
E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288, 2011
Index entries for sequences related to MatulaGoebel numbers


FORMULA

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 tth prime) and t is prime, then a(n)=a(t)+LV(t); if n=p(t) (=the tth 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.


EXAMPLE

a(7)=6 because the rooted tree with MatulaGoebel number 7 is the rooted tree Y (2+2+2=6).
if m>2 then a(2^m) = m(m1) because the rooted tree with MatulaGoebel number 2^m is a star with m edges and the vertices of each of the binom(m,2) pairs of nodes of degree 1 are at distance 2.


MAPLE

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);


CROSSREFS

Cf. A109129, A196048, A196051.
Sequence in context: A175175 A116417 A271410 * A145787 A096111 A101081
Adjacent sequences: A196052 A196053 A196054 * A196056 A196057 A196058


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Sep 29 2011


STATUS

approved



