cp's OEIS Frontend

A152061 Counts of unique periodic binary strings of length n.

%I A152061 #50 Sep 22 2024 02:07:22
%S A152061 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 A152061 4336,32,8194,512,16396,2,33814,2,65536,2054,131074,158,266176,2,
%U A152061 524290,8198,1048816,2,2113462,2,4194316,33272,8388610,2,16842496,128,33555424
%N A152061 Counts of unique periodic binary strings of length n.
%C A152061 a(p) = 2 for p prime.
%H A152061 Alois P. Heinz, <a href="/A152061/b152061.txt">Table of n, a(n) for n = 0..2000</a>
%H A152061 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 A152061 a(n) = 2^n - A001037(n) * n for n>0, a(0) = 0.
%F A152061 a(n) = 2^n - A027375(n) for n>0, a(0) = 0.
%F A152061 a(n) = 2^n - Sum_{d|n} mu(n/d) 2^d for n>0, a(0) = 0.
%F A152061 a(n) = 2^n - A143324(n,2).
%F A152061 a(n) = 2 * A178472(n) for n > 0. - _Alois P. Heinz_, Jul 04 2019
%e A152061 a(3) = 2 = |{ 000, 111 }|, a(4) = 4 = |{ 0000, 1111, 0101, 1010 }|.
%p A152061 with(numtheory):
%p A152061 a:= n-> `if`(n=0, 0, 2^n -add(mobius(n/d)*2^d, d=divisors(n))):
%p A152061 seq(a(n), n=0..100);  # _Alois P. Heinz_, Sep 26 2011
%t A152061 a[0] = 0; a[n_] := 2^n - Sum[MoebiusMu[n/d]*2^d, {d, Divisors[n]}];
%t A152061 Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Jul 04 2019 *)
%o A152061 (Python)
%o A152061 from sympy import mobius, divisors
%o A152061 def A152061(n): return -sum(mobius(n//d)<<d for d in divisors(n,generator=True) if d<n) # _Chai Wah Wu_, Sep 21 2024
%Y A152061 Row sums of A050870.
%Y A152061 A050871 is bisection (even part). - _R. J. Mathar_, Sep 24 2011
%Y A152061 Cf. A008683, A178472.
%K A152061 nonn
%O A152061 0,3
%A A152061 _Jin S. Choi_, Sep 24 2011