# necklaces again

Wouter Meeussen wouter.meeussen at pandora.be
Fri Aug 2 21:05:23 CEST 2002

```A003239 (1,1,2,4,10,26,80,246,810,2704,...) has as comment:
"Also number of necklaces with n beads each of two colors (to get the
correspondence, start at root, walk around outside of tree,
use one
color if move away from the root, the other if towards root)."

in the "necklaces index", it is correctly entered under "balanced"
necklaces.
Wouldn't it be better to change the above comment accordingly?

I find this sequence to correspond with
(sic) necklaces of 2n beads with n white beads (and n black ones). (end-sic)

It's correct if you read ".. of two colors, n beads each, .. "
altough confusing!

Imagine my surprise that

***  a(2n,m)= number of necklaces of 2n beads with 1<=m<=n white ones ***

{1 }
{1, 2},
{1, 3, 4},
{1, 4, 7, 10},
{1, 5, 12, 22, 26},
{1, 6, 19, 43, 66, 80},
{1, 7, 26, 73, 143, 217, 246},
{1, 8, 35, 116, 273, 504, 715, 810},
{1, 9, 46, 172, 476, 1038, 1768, 2438, 2704}
{1, 10, 57, 245, 776, 1944, 3876, 6310, 8398, 9252}
...

is not in EIS.
The row-sums are even beyond SuperSeeker!

since the last entry per row is A003239 :   a(n) = sum {d|n}
(phi(n/d)*C(2d,d))/(2n), n>0. (M. Somos?), it is reasonable to expect an
analogous kind of formula for the triangle above.

Hmm, the next-to-last entry per row looks like A007595. Funny, that.

Any takers?

Wouter Meeussen
wouter.meeussen at pandora.be

```