A208602 Number of n-bead necklaces labeled with numbers -1..1 not allowing reversal, with sum zero.

1, 2, 3, 6, 11, 26, 57, 142, 351, 902, 2333, 6166, 16381, 44046, 119183, 324862, 890291, 2453126, 6789309, 18869426, 52635789, 147325510, 413618615, 1164517198, 3287073461, 9300516890, 26372968983, 74937177538, 213333642443, 608400919106, 1737954608281

Offset: 1

Author

R. H. Hardin, Feb 29 2012

Keywords

• nonn

Examples

			All solutions for n=5:
.-1...-1...-1...-1...-1...-1...-1...-1....0...-1...-1
.-1....1....0...-1....0....0....0....0....0....1...-1
..1...-1....0....1....1...-1....1....0....0....0....0
..1....1....1....0...-1....1....0....0....0....0....1
..0....0....0....1....1....1....0....1....0....0....1
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..100

Crossrefs

Column 1 of A208597.

Programs

Mathematica

comps[r_, m_, k_] := Sum[(-1)^i*Binomial[r - 1 - i*m, k - 1]*Binomial[k, i], {i, 0, Floor[(r - k)/m]}]; a[n_Integer, k_] := DivisorSum[n, EulerPhi[n/#] comps[#*(k + 1), 2 k + 1, #] &]/n; a[n_] = a[n, 1]; Array[a, 31] (* Jean-François Alcover, Nov 01 2017, after Andrew Howroyd *)

Formula

a(n) = (1/n) * Sum_{d | n} totient(n/d) * A002426(d). - Andrew Howroyd, Mar 02 2017