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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 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 STATUS approved

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Last modified October 2 18:23 EDT 2022. Contains 357228 sequences. (Running on oeis4.)