

A198330


MatulaGoebel numbers of central rooted trees.


1



1, 3, 4, 7, 8, 9, 10, 11, 16, 18, 19, 20, 21, 23, 25, 26, 27, 29, 32, 33, 34, 36, 40, 41, 42, 46, 49, 50, 52, 53, 54, 57, 59, 61, 62, 63, 64, 65, 66, 68, 71, 72, 73, 74, 75, 77, 80, 81, 83, 84, 85, 86, 87, 92, 97, 98, 99, 100, 101, 103, 104, 108, 114, 115
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OFFSET

1,2


COMMENTS

A tree is said to be central if its center consists of one point (see the Harary reference, p. 35).


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.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355375, 1979.
J. M. Aldous and R. J. Wilson, Graphs and Applications, Springer, 2000 (p. 179).


LINKS

Table of n, a(n) for n=1..64.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

A198329(n) is the MatulaGoebel number of the rooted tree obtained by removing from the rooted tree with MatulaGoebel number n the vertices of degree one, together with their incident edges. If the repeated application of this pruning operation will not lead to the MatulaGoebel number 2 (corresponding to the 1edge tree), then the starting rooted tree is central. The Maple program is based on this.


EXAMPLE

7 is in the sequence because the rooted tree with MatulaGoebel number 7 is Y which is central. 5 is not in the sequence because the corresponding rooted tree is the pathtree on 4 vertices, a bicentral tree.


MAPLE

with(numtheory): a := proc (n) local r, s, b: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc; b := proc (n) if n = 1 then 1 elif n = 2 then 1 elif bigomega(n) = 1 then ithprime(b(pi(n))) else b(r(n))*b(s(n)) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then b(pi(n)) else b(r(n))*b(s(n)) end if end proc: C := {}: for n to 130 do B := {n}: nn := n: for i while a(nn) > 1 do B := `union`(B, {a(nn)}): nn := a(nn) end do: if member(2, B) = false then C := `union`(C, {n}) else end if end do: C;


CROSSREFS

Cf. A198329, A198331.
Sequence in context: A193406 A104426 A097044 * A232779 A086986 A057811
Adjacent sequences: A198327 A198328 A198329 * A198331 A198332 A198333


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Nov 24 2011


STATUS

approved



