A115124 Number of imprimitive (periodic) 2n-bead black-white reversible complementable necklaces with n black beads.

0, 0, 1, 1, 2, 1, 4, 1, 7, 3, 14, 1, 40, 1, 86, 15, 257, 1, 797, 1, 2523, 87, 8360, 1, 29218, 13, 101341, 765, 361275, 1, 1300415, 1, 4707969, 8361, 17179436, 97, 63097809, 1, 232615772, 101342, 861726044, 1, 3204597995, 1, 11955836263, 1298641, 44748176654, 1

Offset: 0

Author

Valery A. Liskovets, Jan 17 2006

Keywords

• easy
• nonn

Comments

a(p)=1 for prime p.

Programs

Mathematica

b[n_] := (1/(2n)) DivisorSum[n, EulerPhi[n/#] Binomial[2# - 1, # - 1] + EulerPhi[2(n/#)] 2^(# - 1)&];
A006840[n_] := If[n == 0, 1, (b[n] + 2^(n - 2) + Binomial[n - Mod[n, 2], Quotient[n, 2]]/2)/2];
A045633[n_] := If[n==0, 1, Sum[MoebiusMu[n/d] A006840[d], {d, Divisors[n]}] ];
a[n_] := A006840[n] - A045633[n];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Aug 28 2019 *)

Formula

a(n)=A006840(n) - A045633(n).

Extensions

More terms from Jean-François Alcover, Aug 28 2019