

A196068


Visitation length of the rooted tree with MatulaGoebel number n.


2



1, 3, 6, 5, 10, 8, 9, 7, 11, 12, 15, 10, 13, 11, 15, 9, 14, 13, 12, 14, 14, 17, 17, 12, 19, 15, 16, 13, 18, 17, 21, 11, 20, 16, 18, 15, 16, 14, 18, 16, 19, 16, 17, 19, 20, 19, 22, 14, 17, 21, 19, 17, 15, 18, 24, 15, 17, 20, 20, 19, 20, 23, 19, 13, 22, 22, 18, 18, 22, 20, 21, 17, 21, 18, 24, 16, 23, 20, 24, 18
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OFFSET

1,2


COMMENTS

The visitation length of a rooted tree is defined as the sum of the path length and the number of vertices. The path length of a rooted tree is defined as the sum of distances of all vertices to the root of the tree (see the Keijzer et al. reference).
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 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.
M. Keijzer and J. Foster, Crossover bias in genetic programming, Lecture Notes in Computer Sciences, 4445, 2007, 3344.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.


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)=1; if n=p(t) (= the tth prime) then a(n)=a(t)+N(t)+1, where N(t) is the number of nodes of the rooted tree with Matula number t; if n=rs (r,s>=2), then a(n)=a(r)+a(s)1. The Maple program is based on this recursive formula.


EXAMPLE

a(7)=9 because the rooted tree with MatulaGoebel number 7 is the rooted tree Y (1+2+2+4=9).
a(2^m) = 2m+1 because the rooted tree with MatulaGoebel number 2^m is a star with m edges (m+(m+1)=2m+1).


MAPLE

with(numtheory): a := proc (n) local r, s, N: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: N := proc (n) if n = 1 then 1 elif bigomega(n) = 1 then 1+N(pi(n)) else N(r(n))+N(s(n))1 end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then a(pi(n))+N(pi(n))+1 else a(r(n))+a(s(n))1 end if end proc: seq(a(n), n = 1 .. 80);


PROG

(Haskell)
import Data.List (genericIndex)
a196068 n = genericIndex a196068_list (n  1)
a196068_list = 1 : g 2 where
g x = y : g (x + 1) where
y  t > 0 = a196068 t + a061775 t + 1
 otherwise = a196068 r + a196068 s  1
where t = a049084 x; r = a020639 x; s = x `div` r
 Reinhard Zumkeller, Sep 03 2013


CROSSREFS

Cf. A196046, A196047.
Cf. A049084, A020639, A061775.
Sequence in context: A310129 A310130 A298818 * A242239 A123089 A246978
Adjacent sequences: A196065 A196066 A196067 * A196069 A196070 A196071


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Oct 04 2011


STATUS

approved



