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A196051 The Wiener index of the rooted tree with Matula-Goebel number n. 6
0, 1, 4, 4, 10, 10, 9, 9, 20, 20, 20, 18, 18, 18, 35, 16, 18, 31, 16, 32, 32, 35, 31, 28, 56, 31, 48, 29, 32, 50, 35, 25, 56, 32, 52, 44, 28, 28, 50, 46, 31, 46, 29, 52, 72, 48, 50, 40, 48, 75, 52, 46, 25, 64, 84, 42, 46, 50, 32, 67, 44, 56, 67, 36, 76, 76, 28, 48, 72, 70, 46, 59, 46, 44, 102, 42, 79, 68, 52, 62, 88, 50, 48, 62, 79, 46, 75, 71, 40, 92, 71, 67, 84, 72, 71, 54, 75, 65, 104, 96 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph.

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.

REFERENCES

F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.

I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.

I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.

D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288, 2011

Index entries for sequences related to Matula-Goebel numbers

FORMULA

a(1)=0; if n = p(t) (the t-th prime), then a(n)=a(t)+PL(t)+E(t)+1; if n=rs (r,s>=2), then a(n)=a(r)+a(s)+PL(r)E(s)+PL(s)E(r); PL(m) and E(m) denote the path length and the number of edges of the rooted tree with Matula number m (see A196047, A196050). The Maple program is based on this recursive formula.

EXAMPLE

a(7)=9 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (1+1+1+2+2+2=9).

a(2^m) = m^2 because the rooted tree with Matula-Goebel number 2^m is a star with m edges and we have m distances 1 and m(m-1)/2 distances 2; m + m(m-1)=m^2.

MAPLE

with(numtheory): a := proc (n) local r, s, E, PL: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: E := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+E(pi(n)) else E(r(n))+E(s(n)) end if end proc: PL := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+E(pi(n))+PL(pi(n)) else PL(r(n))+PL(s(n)) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then a(pi(n))+PL(pi(n))+1+E(pi(n)) else a(r(n))+a(s(n))+PL(r(n))*E(s(n))+PL(s(n))*E(r(n)) end if end proc: seq(a(n), n = 1 .. 100);

PROG

(Haskell)

import Data.List (genericIndex)

a196051 n = genericIndex a196051_list (n - 1)

a196051_list = 0 : g 2 where

   g x = y : g (x + 1) where

     y | t > 0     = a196051 t + a196047 t + a196050 t + 1

       | otherwise = a196051 r + a196051 s +

                     a196047 r * a196050 s + a196047 s * a196050 r

       where t = a049084 x; r = a020639 x; s = x `div` r

-- Reinhard Zumkeller, Sep 03 2013

CROSSREFS

Cf. A196047, A196050.

Cf. A049084, A020639.

Sequence in context: A050348 A134637 A078910 * A140234 A220044 A219828

Adjacent sequences:  A196048 A196049 A196050 * A196052 A196053 A196054

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Sep 27 2011

STATUS

approved

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Last modified October 23 19:37 EDT 2019. Contains 328373 sequences. (Running on oeis4.)