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A206498
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Number of quasipendant vertices in the rooted tree with Matula-Goebel number n.
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1
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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
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1,2
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COMMENTS
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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 Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel 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 Matula-Goebel numbers of the m branches of T.
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REFERENCES
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R. B. Bapat, Graphs and Matrices, Springer, London, 2010.
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 Y-N. 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.
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LINKS
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FORMULA
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a(1)=0; a(2)=2; if n=p(t) (=the t-th prime) and t is even then a(n)=Lp(t); if n=p(t) (=the t-th 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).
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EXAMPLE
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a(5)=2 because the rooted tree with Matula-Goebel 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.
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MAPLE
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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);
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PROG
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(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
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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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