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A238418 The rounded Randic index of the rooted tree with Matula number n (n.>=2). 0
1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 2, 4, 4, 3, 3, 3, 3, 4, 3, 3, 4, 2, 4, 4, 3, 3, 4, 4, 3, 3, 3, 3, 4, 3, 4, 4, 3, 3, 4, 3, 3, 4, 4, 3, 4, 4, 3, 4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,4

COMMENTS

The Randic index of a tree (known also as branching index) is defined as the summation of 1/(sqrt(d(u)*d(v)) over all edges uv of G, where d(w) denotes the degree of the vertex w.

The Matula 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  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 numbers of the m branches of T.

REFERENCES

M. Randic, On characterization of molecular branching, J. Amer. Chem. Soc., 97, 6609-6615, 1975.

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 Y-N. 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.

E. Deutsch, Rooted tree statistics from Matula numbers, Discrete Applied Math., 160, 2012, 2314-2322.

LINKS

Table of n, a(n) for n=2..100.

E. Deutsch, Rooted tree statistics from Matula numbers, arXiv1111.4288.

Index entries for sequences related to Matula-Goebel numbers

FORMULA

There are recurrence relations that give the Randic index of an "elevated" rooted tree (attach a new vertex to the root which becomes the root of the new tree) and of the merge of two rooted trees (identify the two roots). They make use of the sequence of the degrees of the level-1 vertices (denoted by DL in the Maple program) .

In the Maple program, F(n) gives the actual (not rounded) Randic index of the rooted tree with Matula number n. For example, F(987654321) = 9/10 + 2*sqrt(2) + 7*sqrt(3)/3 + 4*sqrt(5)/5 + sqrt(6)/2 + sqrt(10)/2 + 2*sqrt(15)/15; the corresponding tree is the 29-vertex tree given in Fig. 2 of the Deutsch reference.

EXAMPLE

a(5)=2; indeed the rooted tree with Matula number 5 is the path PQRS (rooted at P). The edges PQ and RS have endpoints of degrees 1 and 2 and the edge QR has endpoints  of degrees 2 and 2; consequently, the contributions of these 3 edges to the Randic index are 1/sqrt(2), 1/sqrt(2), and 1/2, respectively; the Randic index is sqrt(2) + 1/2 = 1.9142.

MAPLE

f := proc (x, y) options operator, arrow: 1/sqrt(x*y) end proc: c := 1; with(numtheory): F := proc (n) local DL, r, s: DL := proc (n) if n = 2 then [1] elif bigomega(n) = 1 then [1+bigomega(pi(n))] else [op(DL(op(1, factorset(n)))), op(DL(n/op(1, factorset(n))))] end if end proc: 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 = 2 then c elif bigomega(n) = 1 then F(pi(n))-(sum(f(DL(pi(n))[j], bigomega(pi(n))), j = 1 .. bigomega(pi(n))))+sum(f(DL(pi(n))[j], 1+bigomega(pi(n))), j = 1 .. bigomega(pi(n)))+f(1, 1+bigomega(pi(n))) else F(r(n))+F(s(n))-(sum(f(DL(r(n))[j], bigomega(r(n))), j = 1 .. bigomega(r(n))))-(sum(f(DL(s(n))[j], bigomega(s(n))), j = 1 .. bigomega(s(n))))+sum(f(DL(r(n))[j], bigomega(n)), j = 1 .. bigomega(r(n)))+sum(f(DL(s(n))[j], bigomega(n)), j = 1 .. bigomega(s(n))) end if end proc: seq(round(F(n)), n = 2 .. 130);

CROSSREFS

Sequence in context: A227533 A235124 A235125 * A085576 A108622 A112348

Adjacent sequences:  A238415 A238416 A238417 * A238419 A238420 A238421

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Mar 16 2014

STATUS

approved

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Last modified October 14 14:06 EDT 2019. Contains 328017 sequences. (Running on oeis4.)