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

0, 0, 0, 1, 2, 6, 12, 27, 54, 113, 228, 465, 934, 1890, 3798, 7644, 15350, 30840, 61878, 124173, 249008, 499318, 1000866, 2005971, 4019446, 8053062, 16131780, 32311665, 64711820, 129589530, 259487040, 519552495, 1040186358, 2082408354

Offset: 0

Author

Henry Bottomley, Dec 22 2000

Keywords

• nonn

Comments

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Illustration of initial terms

Programs

Mathematica

a13[n_] := DivisorSum[n, EulerPhi@(2*#)*2^(n/#)&]/(2*n);
a29[n_] := (1/4)*(Mod[n, 2] + 3)*2^Quotient[n, 2] + DivisorSum[n, EulerPhi[#]*2^(n/#)&]/(2*n);
a[0] = 0; a[n_] := a29[2*n] - a13[2*n] - 2^(n - 1);
Array[a, 34, 0] (* Jean-François Alcover, Nov 05 2017 *)

Formula

a(n) = A059076(2*n) - 2*A059053(2*n).

a(n) = A000029(2*n) - A000013(2*n) - A000079(n-1).

Extensions

More terms from Vladeta Jovovic, Mar 06 2001