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

0,5

Comments

a(p)=1 for prime p.

Links

Table of n, a(n) for n=0..47.

Formula

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

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 *)

Crossrefs

Sequence in context: A252751 A199744 A360870 * A115122 A097360 A360862

Adjacent sequences: A115121 A115122 A115123 * A115125 A115126 A115127

Keyword

easy,nonn

Author

Valery A. Liskovets, Jan 17 2006

Extensions

More terms from Jean-François Alcover, Aug 28 2019

Status

approved