

A198331


MatulaGoebel numbers of bicentral rooted trees.


1



2, 5, 6, 12, 13, 14, 15, 17, 22, 24, 28, 30, 31, 35, 37, 38, 39, 43, 44, 45, 47, 48, 51, 55, 56, 58, 60, 67, 69, 70, 76, 78, 79, 82, 88, 89, 90, 91, 93, 94, 95, 96, 102, 105, 106, 107, 109, 110, 111, 112, 113, 116, 117, 118, 119, 120, 129, 135, 138, 140, 142
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

A tree is said to be bicentral if its center consists of two points (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..61.
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 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

5 is in the sequence because the rooted tree with MatulaGoebel number 5 is the pathtree on 4 vertices which is bicentral.


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: CC := {}: for n to 180 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) = true then CC := `union`(CC, {n}) else end if end do: CC;


CROSSREFS

Cf. A198329, A198330.
Sequence in context: A190120 A069789 A086334 * A057518 A289206 A153485
Adjacent sequences: A198328 A198329 A198330 * A198332 A198333 A198334


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Nov 24 2011


STATUS

approved



