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A005517
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Smallest label f(T) given to a rooted tree T with n nodes in Matula-Goebel labeling.
(Formerly M0706)
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14
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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
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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