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A045656
Number of 2n-bead balanced binary strings, rotationally equivalent to reverse, complement and reversed complement.
8
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
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
KEYWORD
nonn
STATUS
approved