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G.f.: 1 - x/2 - (1/2)*Sum_{d >= 1} (mu(d)/d)*log(1 - 4*x^d) - (1/2)*Sum_{d >= 1} mu(d)*3*x^d/((1 - 2*x^d)*(1 - x^d)) + 3*x*(1 + x + x^2)/((1 + x)*(1 - x)^2). (End)
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A001651[n_] := n - 1 + Ceiling[n/2];
A185172[n_] := If[n==1, 3, Sum[MoebiusMu[d] 4^(n/d), {d, Divisors[n]}]/n];
A038199[n_] := Sum[((2^d-1) MoebiusMu[n/d]), {d, Divisors[n]}];
a[n_] := Switch[n, 0, 1, 1, 3, _, 3 A001651[n] + (1/2)(A185172[n] - 3 * A038199[n])];
a /@ Range[0, 30] (* Jean-François Alcover, Sep 17 2019 *)
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Many mathematicians consider a cyclic composition of n >= 1 with one part or with two parts as achiral by default because the axis of symmetry may pass through the parts. When he defines the DHK transform, Bowers (in the link below) does not accept this convention except possibly for a cyclic composition with two identical (in value and color) parts.
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