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

0,2

Links

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

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

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 *)

Prog

(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//d for d in divisors(n>>(~n&n-1).bit_length(), generator=True)) if n else 1 # Chai Wah Wu, Jul 22 2024

Crossrefs

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

Sequence in context: A363785 A045665 A133492 * A128092 A167569 A032503

Adjacent sequences: A045660 A045661 A045662 * A045664 A045665 A045666

Keyword

nonn

Author

David W. Wilson

Status

approved