A365076 - OEIS

%I #20 Aug 05 2024 01:56:47

%S 2,4,8,12,22,22,44,44,62,64,112,78,158,130,148,172,274,184,344,232,

%T 302,334,508,302,522,472,548,474,814,442,932,684,778,820,904,672,1334,

%U 1030,1100,904,1642,904,1808,1222,1282,1522,2164,1198,2102,1564,1912,1728

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

%C 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.

%H Laurent Vuillon, <a href="https://doi.org/10.36045/bbms/1074791332">Balanced words</a>, Bull. Belg. Math. Soc. Simon Stevin 10(5): 787-805 (December 2003).

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

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

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

%o (Python)

%o from math import gcd

%o def A365076(n): return sum(n//gcd(n,k) for k in range(1,n+1))+1 # _Chai Wah Wu_, Aug 24 2023

%o (Python)

%o from math import prod

%o from sympy import factorint

%o 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

%Y Cf. A057660, A057661.

%K nonn

%O 1,1

%A _Jeffrey Shallit_, Aug 20 2023