A208184 Number of distinct n-colored necklaces with 3 beads per color.

1, 1, 4, 188, 30804, 11211216, 7623616080, 8690922240480, 15391623287043360, 40018220546304026880, 146226577876194816241920, 725283826265926287362419200, 4746982642910487550771226611200, 40045545575592872978305843519334400

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..80

Formula

a(n) = Sum_{d|3} phi(3/d)*(n*d)!/(d!^n*n*3) if n>0 and a(0) = 1.

For n > 0, a(n) = (3*n)!/(3*n*6^n) + 2*(n-1)!/3. - Vaclav Kotesovec, Aug 23 2015

Example

a(0) = 1: the empty necklace.
a(1) = 1: {000}.
a(2) = 4: {000111, 001011, 010011, 010101}.

Maple

with(numtheory);
a:= n-> `if`(n=0, 1, add(phi(3/d) *(n*d)!/(d!^n *3*n), d={1, 3})):
seq(a(n), n=0..20);

Mathematica

Flatten[{1, Table[(3*n)!/(3*n*6^n) + 2*(n-1)!/3, {n, 1, 20}]}] (* Vaclav Kotesovec, Aug 23 2015 *)

Crossrefs

Row n=3 of A208183.

Cf. A000010, A000142.

Sequence in context: A358809 A266492 A285882 * A355613 A172809 A123116

Adjacent sequences: A208181 A208182 A208183 * A208185 A208186 A208187

Keyword

nonn

Author

Alois P. Heinz, Feb 24 2012

Status

approved