A115119 Number of imprimitive (periodic) n-bead necklaces with beads of 2 colors when turning over is allowed.

0, 0, 2, 2, 3, 2, 5, 2, 6, 4, 9, 2, 16, 2, 19, 10, 30, 2, 55, 2, 81, 20, 127, 2, 248, 8, 381, 46, 690, 2, 1302, 2, 2250, 128, 4113, 24, 7896, 2, 14311, 382, 27036, 2, 51641, 2, 96912, 1266, 184411, 2, 354918, 18, 675258, 4114, 1296861, 2, 2501365, 132, 4806102, 14312

Offset

0,3

Comments

a(p)=2 for prime p.

Links

Table of n, a(n) for n=0..57.

Formula

a(n) = A000029(n) - A001371(n).

Mathematica

a29[n_] := a29[n] = (s = If[OddQ[n], 2^((n-1)/2), 2^(n/2-2) + 2^(n/2 - 1)]; a29[0] = 1; Do[s = s + EulerPhi[d] 2^(n/d)/(2n), {d, Divisors[n]}]; s);
a1371[n_] := Sum[MoebiusMu[d] a29[n/d], {d, Divisors[n]}]; a1371[0] = 1;
a[0] = 0; a[n_] := a29[n] - a1371[n];
Array[a, 70, 0] (* Jean-François Alcover, Aug 28 2019 *)

Crossrefs

Cf. A000029, A001371.

Sequence in context: A345268 A164941 A328673 * A066656 A164896 A298422

Adjacent sequences: A115116 A115117 A115118 * A115120 A115121 A115122

Keyword

easy,nonn

Author

Valery A. Liskovets, Jan 17 2006

Extensions

More terms from Jean-François Alcover, Aug 28 2019

Status

approved