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

1,2

Links

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

Formula

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

Example

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

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 *)

Crossrefs

Column 1 of A208597.

Sequence in context: A156807 A032256 A324765 * A051603 A094927 A024423

Adjacent sequences: A208599 A208600 A208601 * A208603 A208604 A208605

Keyword

nonn

Author

R. H. Hardin, Feb 29 2012

Status

approved