A209008 Number of 4-bead necklaces labeled with numbers -n..n not allowing reversal, with sum zero and first and second differences in -n..n.

1, 3, 5, 10, 16, 26, 38, 55, 75, 101, 131, 168, 210, 260, 316, 381, 453, 535, 625, 726, 836, 958, 1090, 1235, 1391, 1561, 1743, 1940, 2150, 2376, 2616, 2873, 3145, 3435, 3741, 4066, 4408, 4770, 5150, 5551, 5971, 6413, 6875, 7360, 7866, 8396, 8948, 9525, 10125

Offset

1,2

Comments

Row 4 of A209007.

Links

R. H. Hardin, Table of n, a(n) for n = 1..210

Formula

Empirical: a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5).

Conjectures from Colin Barker, Jul 07 2018: (Start)

G.f.: x*(1 - 2*x^2 + 3*x^3 - x^4) / ((1 - x)^4*(1 + x)).

a(n) = (4*n^3 + 6*n^2 + 20*n + 48) / 48 for n even.

a(n) = (4*n^3 + 6*n^2 + 20*n + 18) / 48 for n odd.

(End)

Example

Some solutions for n=6:
  -2  0 -3 -3 -1 -3 -2 -3 -2 -3 -1 -2 -1 -2 -1 -2
  -2  0  0 -2  1 -3  1 -1 -1  1 -1  1  0  0  0  2
   2  0  3  3 -1  3  0  3  2  3  1  1 -1  0  0  0
   2  0  0  2  1  3  1  1  1 -1  1  0  2  2  1  0

Crossrefs

Cf. A209007.

Sequence in context: A184800 A350237 A267151 * A032279 A070558 A233758

Adjacent sequences: A209005 A209006 A209007 * A209009 A209010 A209011

Keyword

nonn

Author

R. H. Hardin, Mar 04 2012

Status

approved