A184165 Number of independent (vertex) subsets in the rooted tree with Matula-Goebel number n.

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

Offset: 1

Author

Emeric Deutsch, Oct 19 2011

Keywords

• nonn

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.

Examples

			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.
		

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

Crossrefs

Cf. A049084, A020639, A228731, A228732.

Programs

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

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 *)

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

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=prime(t), then A(n)=[c(t),b(t)+c(t)]; if n=r*s (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