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A060481
Number of orbits of length n in a map whose periodic points come from A059991.
0
1, 0, 1, 0, 3, 2, 9, 0, 28, 24, 93, 20, 315, 288, 1091, 0, 3855, 3626, 13797, 3264, 49929, 47616, 182361, 2720, 671088, 645120, 2485504, 599040, 9256395, 8947294, 34636833, 0, 130150493, 126320640, 490853403, 119302820, 1857283155, 1808400384, 7048151355
OFFSET
1,5
COMMENTS
From Petros Hadjicostas, Jan 15 2018: (Start)
Terms a(2)-a(20) of this sequence and sequence A000048 appear on p. 311 of Sommerville (1909) in the context of sequences of "evolutes" of cyclic compositions of positive integers.
Algebraically, it is easy to prove that this sequence and sequence A000048 have the same odd-indexed terms. (End)
LINKS
V. Chothi, G. Everest, and T. Ward, S-integer dynamical systems: periodic points, J. Reine Angew. Math., 489 (1997), 99-132.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
D. M. Y. Sommerville, On certain periodic properties of cyclic compositions of numbers, Proc. London Math. Soc. S2-7(1) (1909), 263-313.
T. Ward, Almost all S-integer dynamical systems have many periodic points, Ergodic Theory and Dynamical Systems, 18 (1998), 471-486.
FORMULA
If b(n) is the n-th term of A059991, then a(n) = (1/n)* Sum_{d|n} mu(d)*b(n/d). [Corrected by Petros Hadjicostas, Jan 14 2018]
From Petros Hadjicostas, Jan 14 2018: (Start)
a(2*n-1) = A000048(2*n-1) for n >= 1.
a(2^m) = 0 for m >= 1.
G.f.: If B(x) is the g.f. of the sequence b(n) = A059991(n) and C(x) = integrate(B(y)/y, y = 0..x), then the g.f. of the current sequence is A(x) = Sum_{n>=1} (mu(n)/n)*C(x^n). (End)
CROSSREFS
Sequence in context: A329285 A211878 A340461 * A335228 A228492 A340875
KEYWORD
easy,nonn
AUTHOR
EXTENSIONS
a(18)-a(30) by Petros Hadjicostas, Jan 15 2018
STATUS
approved