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

0,5

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

Formula

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

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

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 *)

Crossrefs

Sequence in context: A289443 A029863 A091919 * A335712 A356465 A166963

Adjacent sequences: A059075 A059076 A059077 * A059079 A059080 A059081

Keyword

nonn

Author

Henry Bottomley, Dec 22 2000

Extensions

More terms from Vladeta Jovovic, Mar 06 2001

Status

approved