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A184165 Number of independent (vertex) subsets in the rooted tree with Matula-Goebel number n. 5
2, 3, 5, 5, 8, 8, 9, 9, 13, 13, 13, 14, 14, 14, 21, 17, 14, 22, 17, 23, 23, 21, 22, 26, 34, 22, 35, 24, 23, 36, 21, 33, 34, 23, 37, 40, 26, 26, 36, 43, 22, 38, 24, 37, 57, 35, 36, 50, 41, 59, 37, 38, 33, 62, 55, 44, 43, 36, 23, 66, 40, 34, 61, 65, 58, 58, 26, 41, 57, 62, 43, 76, 38, 40, 93, 44, 60, 60, 37, 83 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A vertex subset in a tree is said to be independent if no pair of vertices is connected by an edge. The empty set is considered to be independent. For example, the 1-edge tree AB has 3 independent subsets: the empty set, {A}, and {B}.

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.

The number of independent subsets of a graph G is called the Merrifield-Simmons index of G.

LINKS

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

M. B. Ahmadi and M. Seif, The Merrifield-Simmons index of an infinite class of dendrimers, Digest J. of Nanomaterials and Biostructures, 5, 2010, 335-338.

É. Czabarka, L. Székely, and S. Wagner, The inverse problem for certain tree parameters, Discrete Appl. Math., 157, 2009, 3314-3319.

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 Rev. 10 (1968) 273.

H. Prodinger and R. F. Tichy, Fibonacci numbers of graphs, Fibonacci Quarterly, 20, 1982, 16-21.

Index entries for sequences related to Matula-Goebel numbers

FORMULA

Define b(n) (c(n)) to be the number of independent subsets of the rooted tree with Matula-Goebel number n that contain (do not contain) the root. We have the following recurrence for the pair A(n)=[b(n),c(n)]. A(1)=[1,1]; if n=p(t) (=the t-th prime), then A(n)=[c(t),b(t)+c(t)]; if n=rs (r,s,>=2), then A(n)=[b(r)b(s), c(r)c(s)]. Clearly, a(n)=b(n)+c(n). See the Czabarka et al. reference (p. 3315, (3)). The Maple program is based on this recursive formula.

a(n) = A228731(n) + A228732(n). - Reinhard Zumkeller, Sep 01 2013

EXAMPLE

a(2)=3 because the tree with the Matula number 2 is the 1-edge tree AB with 3 independent subsets: (empty, {A}, {B}).

a(2655237841)=3216386; the tree D[3] in Fig. 1 of the Ahmadi et al. reference has Merrifield-Simmons index 3216386 (see Table 1). The Matula-Goebel number of D[3] can be found to be 227^4=2655237841.

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 [1, 1] elif bigomega(n) = 1 then [A(pi(n))[2], A(pi(n))[1]+A(pi(n))[2]] else [A(r(n))[1]*A(s(n))[1], A(r(n))[2]*A(s(n))[2]] end if end proc: a := proc (n) options operator, arrow: A(n)[1]+A(n)[2] end proc: seq(a(n), n = 1 .. 80);

MATHEMATICA

r[n_] := FactorInteger[n][[1, 1]];

s[n_] := n/r[n];

A[n_] := A[n] = If[n==1, {1, 1}, If[PrimeOmega[n]==1, {A[PrimePi[n]][[2]], A[PrimePi[n]] // Total}, A[r[n]] * A[s[n]]]];

a[n_] := A[n] // Total;

a /@ Range[1, 80] (* Jean-François Alcover, Sep 20 2019, from Maple *)

PROG

(Haskell)

import Data.List (genericIndex)

a184165 n = a228731 n + a228732 n

a228731 n = genericIndex a228731_list (n - 1)

a228732 n = genericIndex a228732_list (n - 1)

(a228731_list, a228732_list) = unzip $ (1, 1) : map f [2..] where

   f x | i > 0     = (a228732 i, a228731 i + a228732 i)

       | otherwise = (a228731 u * a228731 v, a228732 u * a228732 v)

       where i = a049084 x

             u = a020639 x; v = x `div` u

-- Reinhard Zumkeller, Sep 01 2013

(PARI)

R(n)={my(f=factor(n), g=f); for(i=1, #f~, my([b, c]=R(primepi(f[i, 1]))); f[i, 1]=c; g[i, 1]=b+c); [factorback(f), factorback(g)]}

a(n)=vecsum(R(n)); \\ Andrew Howroyd, Aug 01 2018

CROSSREFS

Cf. A049084, A020639, A228731, A228732.

Sequence in context: A204926 A256663 A188201 * A152771 A325711 A306676

Adjacent sequences:  A184162 A184163 A184164 * A184166 A184167 A184168

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Oct 19 2011

STATUS

approved

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Last modified October 18 18:56 EDT 2019. Contains 328197 sequences. (Running on oeis4.)