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A196068
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Visitation length of the rooted tree with Matula-Goebel number n.
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3
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1, 3, 6, 5, 10, 8, 9, 7, 11, 12, 15, 10, 13, 11, 15, 9, 14, 13, 12, 14, 14, 17, 17, 12, 19, 15, 16, 13, 18, 17, 21, 11, 20, 16, 18, 15, 16, 14, 18, 16, 19, 16, 17, 19, 20, 19, 22, 14, 17, 21, 19, 17, 15, 18, 24, 15, 17, 20, 20, 19, 20, 23, 19, 13, 22, 22, 18, 18, 22, 20, 21, 17, 21, 18, 24, 16, 23, 20, 24, 18
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OFFSET
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1,2
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COMMENTS
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The visitation length of a rooted tree is defined as the sum of the path length and the number of vertices. The path length of a rooted tree is defined as the sum of distances of all vertices to the root of the tree (see the Keijzer et al. reference).
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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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.
M. Keijzer and J. Foster, Crossover bias in genetic programming, Lecture Notes in Computer Sciences, 4445, 2007, 33-44.
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)=1; if n=p(t) (= the t-th prime) then a(n)=a(t)+N(t)+1, where N(t) is the number of nodes of the rooted tree with Matula number t; if n=rs (r,s>=2), then a(n)=a(r)+a(s)-1. The Maple program is based on this recursive formula.
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EXAMPLE
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a(7)=9 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (1+2+2+4=9).
a(2^m) = 2m+1 because the rooted tree with Matula-Goebel number 2^m is a star with m edges (m+(m+1)=2m+1).
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MAPLE
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with(numtheory): a := proc (n) local r, s, N: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: N := proc (n) if n = 1 then 1 elif bigomega(n) = 1 then 1+N(pi(n)) else N(r(n))+N(s(n))-1 end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then a(pi(n))+N(pi(n))+1 else a(r(n))+a(s(n))-1 end if end proc: seq(a(n), n = 1 .. 80);
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PROG
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(Haskell)
import Data.List (genericIndex)
a196068 n = genericIndex a196068_list (n - 1)
a196068_list = 1 : g 2 where
g x = y : g (x + 1) where
y | t > 0 = a196068 t + a061775 t + 1
| otherwise = a196068 r + a196068 s - 1
where t = a049084 x; r = a020639 x; s = x `div` r
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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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