A115123 Number of imprimitive (periodic) bracelets (or necklaces) with n red and blue beads such that the beads switch colors when bracelet is turned over.

0, 1, 1, 2, 1, 5, 1, 9, 4, 22, 1, 63, 1, 156, 24, 469, 1, 1532, 1, 4889, 158, 16546, 1, 57844, 21, 202061, 1480, 720533, 1, 2598406, 1, 9408469, 16548, 34350508, 175, 126167112, 1, 465200334, 202063, 1723346534, 1, 6409076632, 1, 23911272097, 2594970, 89495909410, 1

Offset: 1

Author

Valery A. Liskovets, Jan 17 2006

Keywords

• easy
• nonn

Comments

a(p)=1 for prime p.

Programs

Mathematica

A006080[n_] := Sum[EulerPhi[(n - 1)/k] (Binomial[2k, k]/(2(n - 1))), {k, Divisors[n - 1]}]/2 + 2^(n - 3); A006080[1] = 1;
A066315[n_] := Sum[MoebiusMu[n/d] A006080[d + 1], {d, Divisors[n]}];
a[n_] := A006080[n + 1] - A066315[n];
Array[a, 60] (* Jean-François Alcover, Aug 28 2019 *)

Formula

a(n)=A006080(n+1) - A066315(n).

Extensions

More terms from Jean-François Alcover, Aug 28 2019