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
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
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved