A115120 Number of imprimitive (periodic) 2n-bead black-white reversible necklaces with n black beads.

0, 0, 1, 1, 2, 1, 4, 1, 8, 3, 17, 1, 56, 1, 134, 18, 440, 1, 1434, 1, 4758, 135, 16160, 1, 57254, 16, 200475, 1387, 718152, 1, 2591800, 1, 9398520, 16161, 34324175, 148, 126125330, 1, 465093572, 200476, 1723176740, 1, 6408642359, 1, 23910576236, 2588402, 89494164974, 1, 335923316982, 133, 1264107421202

Offset: 0

Author

Valery A. Liskovets, Jan 17 2006

Keywords

• easy
• nonn

Comments

a(p)=1 for prime p.

Programs

Mathematica

A005648[0] = 1; A005648[n_] := (1/2)(Binomial[2 Quotient[n, 2], Quotient[n, 2]] + DivisorSum[n, EulerPhi[#] Binomial[2n/#, n/#] &]/(2n));
A045628[n_] := If[n == 0, 1, Sum[MoebiusMu[n/d] (2n Binomial[2 Quotient[d, 2], Quotient[d, 2]] + Binomial[2d, d]), {d, Divisors[n]}]/(4n)];
a[n_] := A005648[n] - A045628[n];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Aug 28 2019 *)

Formula

a(n) = A005648(n) - A045628(n).

Extensions

More terms from Jean-François Alcover, Aug 28 2019