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

0,2

Links

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

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)

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A103023 A341732 A111247 * A343256 A129342 A045654

Adjacent sequences: A045653 A045654 A045655 * A045657 A045658 A045659

Keyword

nonn

Author

David W. Wilson

Status

approved