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

1,4

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

Prog

(Python)
p, tm, s = 8, 0, 1
for i in range(p):
    tm += (tm^((1<<s)-1))<<s
    s *= 2
print(1)
for i in range(1, 1<<p):
    a = any(((tm>>(i-j))&((1<<j)-1)) == ((tm>>(i-2*j))&((1<<j)-1)) for j in range(1, i//2+1))
    print(2 if a else 1)
# Andrey Zabolotskiy, Mar 03 2017

Crossrefs

Cf. A090822, A010060.

Sequence in context: A322306 A359778 A305830 * A007061 A001817 A214973

Adjacent sequences: A093911 A093912 A093913 * A093915 A093916 A093917

Keyword

nonn

Author

N. J. A. Sloane, May 26 2004

Status

approved