

A184165


Number of independent (vertex) subsets in the rooted tree with MatulaGoebel 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 1edge tree AB has 3 independent subsets: the empty set, {A}, and {B}.
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.
The number of independent subsets of a graph G is called the MerrifieldSimmons index of G.


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. B. Ahmadi and M. Seif, The MerrifieldSimmons index of an infinite class of dendrimers, Digest J. of Nanomaterials and Biostructures, 5, 2010, 335338.
É. Czabarka, L. Székely, and S. Wagner, The inverse problem for certain tree parameters, Discrete Appl. Math., 157, 2009, 33143319.
F. Goebel, On a 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
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 Rev. 10 (1968) 273.
H. Prodinger and R. F. Tichy, Fibonacci numbers of graphs, Fibonacci Quarterly, 20, 1982, 1621.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

Define b(n) (c(n)) to be the number of independent subsets of the rooted tree with MatulaGoebel 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 tth 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 1edge 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 MerrifieldSimmons index 3216386 (see Table 1). The MatulaGoebel 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] (* JeanFrançois 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



