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A198331
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Matula-Goebel numbers of bicentral rooted trees.
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1
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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
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OFFSET
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1,1
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COMMENTS
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A tree is said to be bicentral if its center consists of two points (see the Harary reference, p. 35).
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REFERENCES
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F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
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, 355-375, 1979.
J. M. Aldous and R. J. Wilson, Graphs and Applications, Springer, 2000 (p. 179).
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LINKS
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FORMULA
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A198329(n) is the Matula-Goebel number of the rooted tree obtained by removing from the rooted tree with Matula-Goebel number n the vertices of degree one, together with their incident edges. If the repeated application of this pruning operation will lead to the Matula-Goebel number 2 (corresponding to the 1-edge tree), then the starting rooted tree is central. The Maple program is based on this.
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EXAMPLE
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5 is in the sequence because the rooted tree with Matula-Goebel number 5 is the path-tree on 4 vertices which is bicentral.
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MAPLE
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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;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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