A208185 Number of distinct n-colored necklaces with 4 beads per color.

1, 1, 10, 2896, 3941598, 15277017432, 135277939358160, 2374127830286012160, 74701932179186551241520, 3911393168902074440088524160, 321715999535364496261149134365440, 39702971502659332476270701578180454400, 7081620512071831837127802029303335215878400

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..50

Formula

a(n) = Sum_{d|4} phi(4/d)*(n*d)!/(d!^n*n*4) if n>0 and a(0) = 1.

For n > 0, a(n) = (4*n)!/(4*n*24^n) + (n-1)!/2 + (2*n-1)!/2^(n+1). - Vaclav Kotesovec, Aug 23 2015

Example

a(0) =  1: the empty necklace.
a(1) =  1: {0000}.
a(2) = 10: {00001111, 00010111, 00100111, 01000111, 00011011, 00110011, 00101011, 01010011, 01001011, 01010101}.

Maple

with(numtheory);
a:= n-> `if`(n=0, 1, add(phi(4/d) *(n*d)!/(d!^n *4*n), d={1, 2, 4})):
seq(a(n), n=0..15);

Mathematica

Flatten[{1, Table[(4*n)!/(4*n*24^n) + (n-1)!/2 + (2*n-1)!/2^(n+1), {n, 1, 15}]}] (* Vaclav Kotesovec, Aug 23 2015 *)

Crossrefs

Row n=4 of A208183.

Cf. A000010, A000142.

Sequence in context: A246116 A132675 A227223 * A123377 A061543 A305666

Adjacent sequences: A208182 A208183 A208184 * A208186 A208187 A208188

Keyword

nonn

Author

Alois P. Heinz, Feb 24 2012

Status

approved