A365076 Number of length-n binary words x such that the infinite word xxxx... is balanced.

2, 4, 8, 12, 22, 22, 44, 44, 62, 64, 112, 78, 158, 130, 148, 172, 274, 184, 344, 232, 302, 334, 508, 302, 522, 472, 548, 474, 814, 442, 932, 684, 778, 820, 904, 672, 1334, 1030, 1100, 904, 1642, 904, 1808, 1222, 1282, 1522, 2164, 1198, 2102, 1564, 1912, 1728

Offset

1,1

Comments

A binary word w is "balanced" if for all lengths and all blocks b of the same length appearing in it, the number of 1's in b can take only two different values. For example, 00111 is not balanced because 00 has no 1's, 01 has one, and 11 has two.

Links

Table of n, a(n) for n=1..52.

Laurent Vuillon, Balanced words, Bull. Belg. Math. Soc. Simon Stevin 10(5): 787-805 (December 2003).

Formula

a(n) = 2*A057661(n).

a(n) = A057660(n) + 1.

Example

For n = 4, the 12 such words are 0000, 0001, 0010, 0100, 0101, 0111 and their bitwise binary complements.

Prog

(Python)
from math import gcd
def A365076(n): return sum(n//gcd(n, k) for k in range(1, n+1))+1 # Chai Wah Wu, Aug 24 2023

Crossrefs

Cf. A057660, A057661.

Sequence in context: A217694 A027677 A103787 * A327480 A330022 A362261

Adjacent sequences: A365073 A365074 A365075 * A365077 A365078 A365079

Keyword

nonn

Author

Jeffrey Shallit, Aug 20 2023

Status

approved