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

%I A115124 #6 Aug 28 2019 08:15:26
%S A115124 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,
%T A115124 13,101341,765,361275,1,1300415,1,4707969,8361,17179436,97,63097809,1,
%U A115124 232615772,101342,861726044,1,3204597995,1,11955836263,1298641,44748176654,1
%N A115124 Number of imprimitive (periodic) 2n-bead black-white reversible complementable necklaces with n black beads.
%C A115124 a(p)=1 for prime p.
%F A115124 a(n)=A006840(n) - A045633(n).
%t A115124 b[n_] := (1/(2n)) DivisorSum[n, EulerPhi[n/#] Binomial[2# - 1, # - 1] + EulerPhi[2(n/#)] 2^(# - 1)&];
%t A115124 A006840[n_] := If[n == 0, 1, (b[n] + 2^(n - 2) + Binomial[n - Mod[n, 2], Quotient[n, 2]]/2)/2];
%t A115124 A045633[n_] := If[n==0, 1, Sum[MoebiusMu[n/d] A006840[d], {d, Divisors[n]}] ];
%t A115124 a[n_] := A006840[n] - A045633[n];
%t A115124 Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Aug 28 2019 *)
%K A115124 easy,nonn
%O A115124 0,5
%A A115124 _Valery A. Liskovets_, Jan 17 2006
%E A115124 More terms from _Jean-François Alcover_, Aug 28 2019