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A365076
Number of length-n binary words x such that the infinite word xxxx... is balanced.
1
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
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
(Python)
from math import prod
from sympy import factorint
def A365076(n): return 1+prod((p**((e<<1)+1)+1)//(p+1) for p, e in factorint(n).items()) # Chai Wah Wu, Aug 05 2024
CROSSREFS
Sequence in context: A217694 A027677 A103787 * A327480 A330022 A362261
KEYWORD
nonn
AUTHOR
Jeffrey Shallit, Aug 20 2023
STATUS
approved