A214310 a(n) is the number of all three-color bracelets (necklaces with turning over allowed) with n beads and the three colors are from a repertoire of n distinct colors, for n >= 3.

1, 24, 180, 1120, 5145, 23016, 91056, 357480, 1327095, 4893680, 17525508, 62254920, 217457695, 753332160, 2581110000, 8779264032, 29624681763, 99350001360, 331159123260, 1098168382080, 3624003213369, 11908069219816, 38972450763000, 127087400895000

Offset: 3

Author

Wolfdieter Lang, Jul 31 2012

Keywords

• nonn

Comments

This is the third column (m=3) of triangle A214306.

Each 3 part partition of n, with the parts written in nonincreasing order, defines a color signature. For a given color signature, say [p[1], p[2], p[3]], with p[1] >= p[2] >= p[3] >= 1, there are A213941(n,k)= A035206(n,k)* A213939(n,k) bracelets if this signature corresponds (with the order of the parts reversed) to the k-th partition of n in Abramowitz-Stegun (A-St) order. See A213941 for more details. Here all p(n,3)= A008284(n,3) partitions of n with 3 parts are considered. The color repertoire for a bracelet with n beads is [c[1], ..., c[n]].

Compare this with A027671 where also single color bracelets are included, and the color repertoire is only [c[1], c[2], c[3]] for all n.

Examples

			a(5) = A213941(5,4) + A213941(5,5) = 60 + 120 = 180 from the bracelet (with colors j for c[j], j=1, 2, ..., 5) 11123 and 11213, both taken cyclically, each representing a class of order A035206(5,4)= 30 (if all 5 colors are used), and 11223, 11232, 12123 and 12213, all taken cyclically, each representing a class of order A035206(5,5)= 30. For example, cyclic(11322) becomes equivalent to cyclic(11223) by turning over or reflection. The multiplicity A035206 depends only on the color signature.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 3..100

Crossrefs

Cf. A213941, A214306, A214307 (m=3, representative bracelets), A214312 (m=4).

Formula

a(n) = A214306(n,3), n >= 3.

a(n) = sum(A213941(n,k), k = A214314(n,3).. (A214314(n,3) - 1 + A008284(n,3))), n >= 3.

a(n) = binomial(n,3) * A056343(n). - Andrew Howroyd, Mar 25 2017

Extensions

a(26) from Andrew Howroyd, Mar 25 2017