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

0,2

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.

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.

Formula

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

Example

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

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 *)

Crossrefs

Antidiagonal sums of A263657.

Cf. A007039, A263655, A263656, A263658.

Sequence in context: A366916 A005855 A096748 * A022866 A350701 A099388

Adjacent sequences: A263656 A263657 A263658 * A263660 A263661 A263662

Keyword

nonn

Author

Felix Fröhlich, Oct 23 2015

Extensions

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

Status

approved