

A196053


The first Zagreb index of the rooted tree with MatulaGoebel 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 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 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

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 MatulaGoebel numbers


FORMULA

a(1)=0; if n = p(t) (the tth prime), then a(n)=a(t)+2+2G(t); if n=rs (r,s>=2), then a(n)=a(r)+a(s)G(r)^2G(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 MatulaGoebel number 7 is the rooted tree Y (1+9+1+1=12).
a(2^m) = m(m+1) because the rooted tree with MatulaGoebel 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))^2bigomega(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



