Re: problem with a necklace sequence


mensanator@xxxxxxxxxxx wrote:
lloyd wrote:
A sequence came up in a puzzle I was working on, that appears to agree
with A066313 in Sloane's online encyclopedia of integer sequences. My
context was completely unrelated to the one that appears in the OEIS.
In trying to understand the correspondence I realised I didn't
understand the definition of A066313. Can anyone help? It says:

"Number of aperiodic bracelets (or necklaces) with n red or blue beads
such that the beads switch colors when bracelet is turned over."

The sequence starts (with n=1):

1, 1, 1, 2, 3, 6, 9, 18, 28, 57, 93, 181, 315 ...

here's an example, as far as I understand it, for n=6:

r r b b r b "turned over" becomes (read starting from the same bead)
r b r b b r which is a rotation of
b b r r b r which is the original with the colours switched.

These two, along with rrrbbb, must be the three cases the OEIS means
for
n=6, as far as I can tell.

Count again. It's 6 for n=6.

I too can find only these three - though it's unclear if rrbbrb should
be considered the same necklace as bbrrbr (being its mirror image, i.e.
the same when turned over). If it is then there seem to be only two
possibilities. And FWIW I can't make any sense of the OEIS explanation
either.

Did you actually find 6, or did you just take that number from the OEIS
sequence?



But here's my question: how can you ever have such a necklace for odd
n?
For red and blue to switch roles there must be the same number of each,

no? A066313 has positive values for all n though, so I must be missing
something. Can anyone tell me what?

Thanks --Lloyd

.

Relevant Pages

  • Re: problem with a necklace sequence
    ... with A066313 in Sloane's online encyclopedia of integer sequences. ... context was completely unrelated to the one that appears in the OEIS. ... "Number of aperiodic bracelets with n red or blue beads ... For red and blue to switch roles there must be the same number of each, ...
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  • Re: problem with a necklace sequence
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  • Re: problem with a necklace sequence
    ... context was completely unrelated to the one that appears in the OEIS. ... "Number of aperiodic bracelets with n red or blue beads ... The sequence starts: ... For red and blue to switch roles there must be the same number of each, ...
    (sci.math)
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