A093914 a(1) = 1; for n > 1, a(n) = curling number of (b(1),...,b(n-1)), where b() = Thue-Morse sequence A010060 (with offset changed to 1).

1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2

Offset: 1

Author

N. J. A. Sloane, May 26 2004

Keywords

• nonn

Comments

The curling number of a finite string S = (s(1),...,s(n)) is the largest integer k such that S can be written as xy^k for strings x and y (where y has positive length).

From Andrey Zabolotskiy, Mar 03 2017: (Start)

The sequence consists of 1's and 2's only.

If 2^k>=n-1, then a(n+2^k)>=a(n).

The density of 1's seems to converge to 1/6.

(End)

Links

• Andrey Zabolotskiy, Table of n, a(n) for n = 1..16384
• F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence, J. Integer Sequences, Vol. 10 (2007), #07.1.2.
• F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence [pdf, ps].
• Index entries for sequences related to curling numbers

Crossrefs

Cf. A090822, A094840, A010060.

Programs

Mathematica

(* Function curlN is defined in A094840 *)
(* Function ThueMorse needs Mma version >= 11 *)
a[n_] := If[n == 1, 1, curlN[Array[ThueMorse, n-1, 0]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 18 2024 *)

Python

p, tm, s = 8, 0, 1
for i in range(p):
    tm += (tm^((1<>(i-j))&((1<>(i-2*j))&((1<Andrey Zabolotskiy, Mar 03 2017