

A196054


The second Zagreb index of the rooted tree with MatulaGoebel number n.


2



0, 1, 4, 4, 8, 8, 9, 9, 12, 12, 12, 14, 14, 14, 16, 16, 14, 19, 16, 18, 18, 16, 19, 22, 20, 19, 24, 21, 18, 23, 16, 25, 20, 18, 22, 28, 22, 22, 23, 26, 19, 26, 21, 22, 28, 24, 23, 32, 24, 27, 22, 26, 25, 34, 24, 30, 26, 23, 18, 32, 28, 20, 31, 36, 27, 27, 22, 24, 28, 30, 26, 39, 26, 28, 32, 30, 26, 31, 22, 36, 40, 23, 24, 36, 26, 26, 27, 30, 32, 38
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


COMMENTS

The second Zagreb index of a simple connected graph g is the sum of the degree products d(i)d(j) over all edges ij of g.
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

I. Gutman and K. C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50, 2004, 8392.
F. Goebel, On a 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and K.C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50, 2004, 8392.
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.
S. Nikolic, G. Kovacevic, A. Milicevic, and N. Trinajstic, The Zagreb indices 30 years after, Croatica Chemica Acta, 76, 2003, 113124.


LINKS

Table of n, a(n) for n=1..90.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

a(1)=0; if n=p(t) (the tth prime), then a(n) = a(t)+b(t)+G(t)+1; if n=rs (r,s>=2), then a(n)=a(r)+a(s)+b(r)G(s)+b(s)G(r); here b(m) is the sum of the degrees of the nodes at level 1 of the rooted tree having MatulaGoebel number m and 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)=9 because the rooted tree with MatulaGoebel number 7 is the rooted tree Y (1*3+3*1+3*1=9).
a(2^m) = m^2 because the rooted tree with MatulaGoebel number 2^m is a star with m edges.


MAPLE

with(numtheory): a := proc (n) local r, s, b: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: b := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+bigomega(pi(n)) else b(r(n))+b(s(n)) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then a(pi(n))+b(pi(n))+bigomega(pi(n))+1 else a(r(n))+a(s(n))+b(r(n))*bigomega(s(n))+b(s(n))*bigomega(r(n)) end if end proc: seq(a(n), n = 1 .. 90);


CROSSREFS

Cf. A196052, A196053.
Sequence in context: A246066 A076359 A105675 * A292135 A053249 A071339
Adjacent sequences: A196051 A196052 A196053 * A196055 A196056 A196057


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Sep 28 2011


STATUS

approved



