A263659 Number of (0, 1)-necklaces of length n without zigzags (see reference for precise definition).

0, 2, 2, 2, 3, 4, 5, 6, 8, 10, 15, 20, 31, 42, 64, 94, 143, 212, 329, 494, 766, 1170, 1811, 2788, 4341, 6714, 10462, 16274, 25415, 39652, 62075, 97110, 152288, 238838, 375167, 589528, 927555, 1459962, 2300348, 3626242, 5721045

Offset: 0

Author

Felix Fröhlich, Oct 23 2015

Keywords

• nonn

Comments

See page 16 in the reference.

A zigzag is a substring which is either 010 or 101. The necklaces 01 and 10 are considered to be with a zigzag. Necklaces do not allow turnover.

Examples

			For n=5 the necklaces are 00000, 11111, 00011, 00111 so a(5)=4.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200
• E. Munarini and N. Z. Salvi, Circular Binary Strings without Zigzags, Integers: Electronic Journal of Combinatorial Number Theory 3 (2003), #A19.

Crossrefs

Antidiagonal sums of A263657.

Cf. A007039, A263655, A263656, A263658.

Programs

Mathematica

(* b = A007039 *) b[n_ /; n<4] = 2; b[4] = 6; b[n_] := b[n] = 2*b[n-1] - b[n-2] + b[n-4];
a[0] = 0; a[n_] := (1/n) * DivisorSum[n, EulerPhi[n/#] * b[#]&];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *)

Formula

a(n) = (1/n) * Sum_{d | n} totient(n/d) * A007039(d). - Andrew Howroyd, Feb 26 2017

Extensions

a(25)-a(40) from Andrew Howroyd, Feb 26 2017