A045656 Number of 2n-bead balanced binary strings, rotationally equivalent to reverse, complement and reversed complement.

1, 2, 6, 8, 22, 32, 48, 100, 150, 260, 336, 684, 784, 1640, 1868, 3728, 4246, 8672, 9372, 19420, 20752, 42736, 45700, 94164, 98832, 204632, 214584, 441764, 460524, 950216, 985968, 2031556, 2101398, 4323888, 4465056, 9174400, 9444988

Offset: 0

Author

David W. Wilson

Keywords

• nonn

Links

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

Crossrefs

Cf. A000984, A023900, A045665, A045653, A045654, A045655, A045674.

Programs

Mathematica

b[n_] := Module[{t = 0, r = n}, If[n == 0, 1,  While[Mod[r, 2] == 0, r = r/2; t += 2^(r - 1)]; t + 2^Quotient[r, 2]]];
c[n_] := Sum[MoebiusMu[d]*d, {d, Divisors[n]}];
a[n_] := If[n == 0, 1, 2*Sum[c[n/d]*d*b[d], {d, Divisors[n]}]];
a /@ Range[0, 36] (* Jean-François Alcover, Sep 23 2019, from PARI *)

PARI

\\ here b(n) is A045674,  c(n) is A023900.
b(n) = if(n<1, n==0, my(t=0, r=n); while(r%2==0, r=r/2; t+=2^(r-1)); t + 2^(r\2));
c(n) = {sumdiv(n,d, moebius(d)*d)}
a(n) = if(n<1, n==0, 2*sumdiv(n, d, c(n/d)*d*b(d))); \\ Andrew Howroyd, Sep 15 2019

Formula

From Andrew Howroyd, Sep 15 2019: (Start)

Inverse Moebius transform of A045665.

a(n) = 2*Sum_{d|n} A023900(n/d)*d*A045674(d) for n > 0. (End)