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%I #46 Oct 26 2022 13:11:42
%S 0,0,2,2,4,2,10,2,16,8,34,2,76,2,130,38,256,2,568,2,1036,134,2050,2,
%T 4336,32,8194,512,16396,2,33814,2,65536,2054,131074,158,266176,2,
%U 524290,8198,1048816,2,2113462,2,4194316,33272,8388610,2,16842496,128,33555424
%N Counts of unique periodic binary strings of length n.
%C a(p) = 2 for p prime.
%H Alois P. Heinz, <a href="/A152061/b152061.txt">Table of n, a(n) for n = 0..2000</a>
%H Achilles A. Beros, Bjørn Kjos-Hanssen, and Daylan Kaui Yogi, <a href="https://arxiv.org/abs/1902.00812">Planar digraphs for automatic complexity</a>, arXiv:1902.00812 [cs.FL], 2019.
%F a(n) = 2^n - A001037(n) * n for n>0, a(0) = 0.
%F a(n) = 2^n - A027375(n) for n>0, a(0) = 0.
%F a(n) = 2^n - Sum_{d|n} mu(n/d) 2^d for n>0, a(0) = 0.
%F a(n) = 2^n - A143324(n,2).
%F a(n) = 2 * A178472(n) for n > 0. - _Alois P. Heinz_, Jul 04 2019
%e a(3) = 2 = |{ 000, 111 }|, a(4) = 4 = |{ 0000, 1111, 0101, 1010 }|.
%p with(numtheory):
%p a:= n-> `if`(n=0, 0, 2^n -add(mobius(n/d)*2^d, d=divisors(n))):
%p seq(a(n), n=0..100); # _Alois P. Heinz_, Sep 26 2011
%t a[0] = 0; a[n_] := 2^n - Sum[MoebiusMu[n/d]*2^d, {d, Divisors[n]}];
%t Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Jul 04 2019 *)
%Y Row sums of A050870.
%Y A050871 is bisection (even part). - _R. J. Mathar_, Sep 24 2011
%Y Cf. A008683, A178472.
%K nonn
%O 0,3
%A _Jin S. Choi_, Sep 24 2011
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