%I #11 Sep 23 2019 09:45:00

%S 1,2,6,8,22,32,48,100,150,260,336,684,784,1640,1868,3728,4246,8672,

%T 9372,19420,20752,42736,45700,94164,98832,204632,214584,441764,460524,

%U 950216,985968,2031556,2101398,4323888,4465056,9174400,9444988

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

%H Andrew Howroyd, <a href="/A045656/b045656.txt">Table of n, a(n) for n = 0..1000</a>

%F From _Andrew Howroyd_, Sep 15 2019: (Start)

%F Inverse Moebius transform of A045665.

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

%t 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]]];

%t c[n_] := Sum[MoebiusMu[d]*d, {d, Divisors[n]}];

%t a[n_] := If[n == 0, 1, 2*Sum[c[n/d]*d*b[d], {d, Divisors[n]}]];

%t a /@ Range[0, 36] (* _Jean-François Alcover_, Sep 23 2019, from PARI *)

%o (PARI) \\ here b(n) is A045674, c(n) is A023900.

%o 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));

%o c(n) = {sumdiv(n,d, moebius(d)*d)}

%o a(n) = if(n<1, n==0, 2*sumdiv(n, d, c(n/d)*d*b(d))); \\ _Andrew Howroyd_, Sep 15 2019

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

%K nonn

%O 0,2

%A _David W. Wilson_