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A059078 Number of orientable necklaces with 2n beads and two colors which when turned over produce their own color complement.

%I A059078 #16 Dec 14 2017 21:15:49
%S A059078 0,0,0,1,2,6,12,27,54,113,228,465,934,1890,3798,7644,15350,30840,
%T A059078 61878,124173,249008,499318,1000866,2005971,4019446,8053062,16131780,
%U A059078 32311665,64711820,129589530,259487040,519552495,1040186358,2082408354
%N A059078 Number of orientable necklaces with 2n beads and two colors which when turned over produce their own color complement.
%C A059078 Clearly in each necklace the number of beads of each of the two colors must be equal and so the number of beads must be even, if a(n) is to be positive.
%H A059078 G. C. Greubel, <a href="/A059078/b059078.txt">Table of n, a(n) for n = 0..1000</a>
%H A059078 <a href="/A059078/a059078.gif">Illustration of initial terms</a>
%F A059078 a(n) = A059076(2*n) - 2*A059053(2*n).
%F A059078 a(n) = A000029(2*n) - A000013(2*n) - A000079(n-1).
%t A059078 a13[n_] := DivisorSum[n, EulerPhi@(2*#)*2^(n/#)&]/(2*n);
%t A059078 a29[n_] := (1/4)*(Mod[n, 2] + 3)*2^Quotient[n, 2] + DivisorSum[n, EulerPhi[#]*2^(n/#)&]/(2*n);
%t A059078 a[0] = 0; a[n_] := a29[2*n] - a13[2*n] - 2^(n - 1);
%t A059078 Array[a, 34, 0] (* _Jean-François Alcover_, Nov 05 2017 *)
%K A059078 nonn
%O A059078 0,5
%A A059078 _Henry Bottomley_, Dec 22 2000
%E A059078 More terms from _Vladeta Jovovic_, Mar 06 2001