A060481 Number of orbits of length n in a map whose periodic points come from A059991.

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

Author

Thomas Ward

Keywords

• easy
• nonn

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.

Crossrefs

Cf. A000048, A059991.

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)

Extensions

a(18)-a(30) by Petros Hadjicostas, Jan 15 2018