

A206498


Number of quasipendant vertices in the rooted tree with MatulaGoebel number n.


1



0, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 3, 1, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 3, 2, 1, 2, 2, 2, 3, 2, 2, 3, 2, 3, 3, 2, 2, 3, 3, 3, 2, 2, 3, 2, 3, 1, 4, 2, 2, 2, 3, 2, 3, 3, 2, 3, 1, 3, 3, 2, 2, 3, 3, 2, 3, 3, 3, 3, 2, 2, 4, 2, 2, 4, 3, 3, 3, 2, 3, 3, 2, 2, 4, 3, 3, 2, 3, 2, 2, 3, 3, 3, 3, 3, 3, 4, 3, 3, 2, 2, 4, 3, 3, 3, 2, 3, 3, 3, 3, 4, 2, 2, 3
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OFFSET

1,2


COMMENTS

A pendant vertex in a tree is a vertex having degree 1. A vertex is called quasipendant if it is adjacent to a pendant vertex (see the Bapat reference, p. 106).
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

R. B. Bapat, Graphs and Matrices, Springer, London, 2010.
F. Goebel, On a 11 correspondence 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 YN. 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 Review, 10, 1968, 273.


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

a(1)=0; a(2)=2; if n=p(t) (=the tth prime) and t is even then a(n)=Lp(t); if n=p(t) (=the tth prime) and t>=3 is odd, then a(n) = 1+Lp(t); if n is composite, then a(n)=Lp(n); here Lp stands for "number of leaf parents" (see A196062).


EXAMPLE

a(5)=2 because the rooted tree with MatulaGoebel number 5 is the path tree A  B  C  D; the pendant vertices are A and D while the quasipendant ones are B and C.


MAPLE

with(numtheory): Lp := 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 0 elif n = 2 then 1 elif bigomega(n) = 1 then Lp(pi(n)) elif `mod`(r(n), 2) = 0 and `mod`(s(n), 2) = 0 then Lp(r(n))+Lp(s(n))1 else Lp(r(n))+Lp(s(n)) end if end proc: a := proc (n) if n = 1 then 0 elif n = 2 then 2 elif bigomega(n) = 1 and 0 < `mod`(pi(n), 2) then 1+Lp(pi(n)) elif bigomega(n) = 1 then Lp(pi(n)) else Lp(n) end if end proc: seq(a(n), n = 1 .. 120);


PROG

(Haskell)
a206498 1 = 0
a206498 2 = 2
a206498 x = if t > 0 then a196062 t + t `mod` 2 else a196062 x
where t = a049084 x
 Reinhard Zumkeller, Sep 03 2013


CROSSREFS

Cf. A196062.
Cf. A049084.
Sequence in context: A143209 A163994 A156593 * A184848 A184720 A054526
Adjacent sequences: A206495 A206496 A206497 * A206499 A206500 A206501


KEYWORD

nonn


AUTHOR

Emeric Deutsch, May 22 2012


STATUS

approved



