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A196053 The first Zagreb index of the rooted tree with Matula-Goebel number n. 3
0, 2, 6, 6, 10, 10, 12, 12, 14, 14, 14, 16, 16, 16, 18, 20, 16, 20, 20, 20, 20, 18, 20, 24, 22, 20, 24, 22, 20, 24, 18, 30, 22, 20, 24, 28, 24, 24, 24, 28, 20, 26, 22, 24, 28, 24, 24, 34, 26, 28, 24, 26, 30, 32, 26, 30, 28, 24, 20, 32, 28, 22, 30, 42, 28, 28, 24, 26, 28, 30, 28, 38, 26, 28, 32, 30, 28, 30, 24, 38, 36, 24, 24, 34, 28, 26, 28, 32, 34, 36, 30, 30, 26, 28, 32, 46, 28, 32, 32, 36 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices.

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 K. C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50, 2004, 83-92.

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.

S. Nikolic, G. Kovacevic, A. Milicevic, and N. Trinajstic, The Zagreb indices 30 years after, Croatica Chemica Acta, 76, 2003, 113-124.

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)+2+2G(t); if n=rs (r,s>=2), then a(n)=a(r)+a(s)-G(r)^2-G(s)^2+G(n)^2;  G(m) is the number of prime factors of m, counted with multiplicities. The Maple program is based on this recursive formula.

EXAMPLE

a(7)=12 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (1+9+1+1=12).

a(2^m) = m(m+1) because the rooted tree with Matula-Goebel number 2^m is a star with m edges.

MAPLE

with(numtheory): a := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 0 elif bigomega(n) = 1 then a(pi(n))+2+2*bigomega(pi(n)) else a(r(n))+a(s(n))-bigomega(r(n))^2-bigomega(s(n))^2+bigomega(n)^2 end if end proc: seq(a(n), n = 1 .. 100);

PROG

(Haskell)

import Data.List (genericIndex)

a196053 n = genericIndex a196053_list (n - 1)

a196053_list = 0 : g 2 where

   g x = y : g (x + 1) where

     y | t > 0     = a196053 t + 2 + 2 * a001222 t

       | otherwise = a196053 r + a196053 s -

                     a001222 r ^ 2  - a001222 s ^ 2 + a001222 x ^ 2

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

-- Reinhard Zumkeller, Sep 03 2013

CROSSREFS

Cf. A196054.

Cf. A049084, A020639, A001222.

Sequence in context: A021978 A141377 A006955 * A063210 A114718 A102261

Adjacent sequences:  A196050 A196051 A196052 * A196054 A196055 A196056

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Sep 28 2011

STATUS

approved

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Last modified January 22 10:32 EST 2019. Contains 319363 sequences. (Running on oeis4.)