A261497 Number of necklaces with n white beads and 3*n black beads.

1, 1, 4, 19, 116, 776, 5620, 42288, 328756, 2615104, 21191904, 174303163, 1451430692, 12211799224, 103655949072, 886568153744, 7633233556276, 66105170315084, 575445692499952, 5032380942945322, 44191451788248416, 389514699013012242, 3444925385336301684

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.

F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]

Eric Weisstein's World of Mathematics, Necklace

Wikipedia, Necklace (combinatorics)

Index entries for sequences related to necklaces

Formula

a(n) = 1/(4*n) * Sum_{d|n} C(4*n/d,n/d) * A000010(d) for n>0, a(0) = 1.

a(n) ~ 2^(8*n-3/2) / (3^(3*n) * sqrt(3*Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 22 2015

Maple

with(numtheory):
a:= n-> `if`(n=0, 1, add(binomial(4*n/d, n/d)
             *phi(d), d=divisors(n))/(4*n)):
seq(a(n), n=0..25);

Crossrefs

Column k=3 of A261494.

Sequence in context: A186359 A203268 A209673 * A217989 A281817 A060907

Adjacent sequences: A261494 A261495 A261496 * A261498 A261499 A261500

Keyword

nonn

Author

Alois P. Heinz, Aug 21 2015

Status

approved