A045663 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to complement.

1, 2, 4, 6, 16, 30, 60, 126, 256, 504, 1020, 2046, 4080, 8190, 16380, 32730, 65536, 131070, 262080, 524286, 1048560, 2097018, 4194300, 8388606, 16776960, 33554400, 67108860, 134217216, 268435440, 536870910, 1073740740, 2147483646

Offset: 0

Author

David W. Wilson

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000048, A007727, A045662, A045664, A064355.

Programs

Mathematica

a[n_] := If[n==0, 1, 2n Total[MoebiusMu[#]*2^(n/#)& /@ Select[Divisors[n], OddQ]]/(2n)];
a /@ Range[0, 31] (* Jean-François Alcover, Sep 23 2019 *)

PARI

a(n)={if(n<1, n==0, sumdiv(n, d, if(d%2, moebius(d)*2^(n/d))))} \\ Andrew Howroyd, Sep 14 2019

Python

from sympy import mobius, divisors
def A045663(n): return sum(mobius(d)<>(~n&n-1).bit_length(),generator=True)) if n else 1 # Chai Wah Wu, Jul 22 2024

Formula

a(n) = 2*n*A000048(n) = n*A064355(n) for n > 0.

a(n) = Sum{d|n, d odd} mu(d) * 2^(n/d) for n > 0. - Andrew Howroyd, Sep 14 2019