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A005517 Smallest label f(T) given to a rooted tree T with n nodes in Matula-Goebel labeling.
(Formerly M0706)
11
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.

REFERENCES

I. Gutman and A. Ivic, Graphs with maximal and minimal Matula numbers, Bulletin CVII Acad. Serbe, Sciences Math., 107, No. 19, 1994, 65-74.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=1..34.

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.

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.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Index entries for sequences related to Matula-Goebel numbers

Index entries for sequences related to rooted trees

Index entries for sequences related to trees

Index entries for linear recurrences with constant coefficients, signature (0, 0, 5).

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

Adjacent sequences:  A005514 A005515 A005516 * A005518 A005519 A005520

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified January 15 20:47 EST 2019. Contains 319184 sequences. (Running on oeis4.)