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A152061 Counts of unique periodic binary strings of length n. 2
0, 0, 2, 2, 4, 2, 10, 2, 16, 8, 34, 2, 76, 2, 130, 38, 256, 2, 568, 2, 1036, 134, 2050, 2, 4336, 32, 8194, 512, 16396, 2, 33814, 2, 65536, 2054, 131074, 158, 266176, 2, 524290, 8198, 1048816, 2, 2113462, 2, 4194316, 33272, 8388610, 2, 16842496, 128, 33555424 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(p) = 2 for p prime.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Achilles A. Beros, Bjørn Kjos-Hanssen, Daylan Kaui, The number of long words having a given automatic complexity*, 2018.

FORMULA

a(n) = 2^n - A001037(n) * n for n>0, a(0) = 0.

a(n) = 2^n - A027375(n) for n>0, a(0) = 0.

a(n) = 2^n - Sum_{d|n} mu(n/d) 2^d for n>0, a(0) = 0.

a(n) = 2^n - A143324(n,2).

EXAMPLE

a(3) = 2 = |{ 000, 111 }|, a(4) = 4 = |{ 0000, 1111, 0101, 1010 }|.

MAPLE

with(numtheory):

a:= n-> `if`(n=0, 0, 2^n -add(mobius(n/d)*2^d, d=divisors(n))):

seq(a(n), n=0..100);  # Alois P. Heinz, Sep 26 2011

CROSSREFS

Row sums of A050870.

A050871 is bisection (even part). - R. J. Mathar, Sep 24 2011

Cf. A008683.

Sequence in context: A076078 A292786 A053204 * A103314 A306019 A194560

Adjacent sequences:  A152058 A152059 A152060 * A152062 A152063 A152064

KEYWORD

nonn

AUTHOR

Jin S. Choi, Sep 24 2011

STATUS

approved

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Last modified May 25 01:59 EDT 2019. Contains 323534 sequences. (Running on oeis4.)