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A005517 Smallest label f(T) given to a rooted tree T with n nodes in Matula-Goebel labeling.
(Formerly M0706)
14
1, 2, 3, 5, 9, 15, 25, 45, 75, 125, 225, 375, 625, 1125, 1875, 3125, 5625, 9375, 15625, 28125, 46875, 78125, 140625, 234375, 390625, 703125, 1171875, 1953125, 3515625, 5859375, 9765625, 17578125, 29296875, 48828125 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Let p(1)=2, ... denote the primes. The label f(T) for a rooted tree T is 1 if T has 1 node, otherwise f(T) = Product p(f(T_i)) where the T_i are the subtrees obtained by deleting the root and the edges adjacent to it.
For n >= 3, this is also the minimum number of Hamiltonian paths in a strong tournament with n vertices (Busch). - Gordon Royle, Jan 24 2022
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Arthur H. Busch, A Note on the Number of Hamiltonian Paths in Strong Tournaments, Electronic Journal of Combinatorics, N3 (2006).
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, Graphs with maximal and minimal Matula numbers, Bulletin CVII Acad. Serbe, Sciences Math., 107, No. 19, 1994, 65-74.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
FORMULA
a(1)=1; a(2)=2; a(n) = 3*5^((n-3)/3) if n=0 (mod 3); a(n)=5^((n-1)/3) if n>=4 and n=1 (mod 3); a(n)=9*5^((n-5)/3) if n>=5 and n = 2 (mod 3) (see the Gutman and Ivic 1994 paper). - Emeric Deutsch, Apr 15 2012
G.f.: z*(1+2*z+3*z^2-z^4)/(1-5*z^3) (conjectured by Simon Plouffe).
a(n+3) = 5*a(n) for n >= 3 under plausible assumptions about growth of prime numbers. - David W. Wilson, Jul 05 2001
A091233(n) = (A005518(n)-a(n))+1. - Antti Karttunen, May 24 2004
MAPLE
a := proc (n) if n = 1 then 1 elif n = 2 then 2 elif `mod`(n, 3) = 0 then 3*5^((1/3)*n-1) elif `mod`(n, 3) = 1 then 5^((1/3)*n-1/3) else 9*5^((1/3)*n-5/3) end if end proc: seq(a(n), n = 1 .. 34); # Emeric Deutsch, Apr 15 2012
A005517:=(-1-2*z-3*z**2+z**4)/(-1+5*z**3); # conjectured by Simon Plouffe in his 1992 dissertation
MATHEMATICA
Join[{1, 2}, LinearRecurrence[{0, 0, 5}, {3, 5, 9}, 40]] (* Harvey P. Dale, Feb 25 2012 *)
a[n_] := Which[n == 1, 1, n == 2, 2, Mod[n, 3] == 0, 3*5^((1/3)*n-1), Mod[n, 3] == 1, 5^((1/3)*n-1/3), True, 9*5^((1/3)*n-5/3)]; Table[a[n], {n, 1, 34}] (* Jean-François Alcover, Mar 06 2014, after Emeric Deutsch *)
CROSSREFS
Cf. A061773. See A005518 for the largest value of f(T).
Sequence in context: A017989 A017990 A228646 * A034063 A034073 A114623
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)