A115122 Number of imprimitive (periodic) 2n-bead black-white complementable necklaces with n black beads.

0, 0, 1, 1, 2, 1, 4, 1, 7, 3, 16, 1, 49, 1, 129, 17, 415, 1, 1408, 1, 4659, 130, 16081, 1, 56858, 15, 200171, 1367, 716733, 1, 2590529, 1, 9393119, 16082, 34319668, 142, 126104312, 1, 465076161, 200172, 1723097474, 1, 6408573532, 1, 23910271229, 2587214, 89493903439

Offset: 0

Author

Valery A. Liskovets, Jan 17 2006

Keywords

• easy
• nonn

Comments

a(p)=1 for prime p.

Programs

Mathematica

A045629[n_] := If[n==0, 1, (1/(2n)) DivisorSum[n, EulerPhi[n/#] Binomial[ 2 # - 1, # - 1] + EulerPhi[2 (n/#)] 2^(# - 1)&]];
A045632[n_] := If[n==0, 1, Sum[MoebiusMu[n/d] A045629[d], {d, Divisors[n]}]];
a[n_] := A045629[n] - A045632[n];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Aug 28 2019 *)

Formula

a(n)=A045629(n) - A045632(n).

Extensions

More terms from Jean-François Alcover, Aug 28 2019