Re: problem with a necklace sequence

jaapsch 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.

It is a very misleading description in the OEIS. When I looked at:
http://www.research.att.com/~njas/sequences/A053656
it began to make a bit more sense.

Let's look again at the 3 necklaces you mention:
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.

All these three are to be considered identical. Now how many aperiodic
necklaces are there if necklaces that differ by rotation and/or by
(flip-over + swap colours) are considered identical?

Let's list them all:
r r r r r r (=b b b b b b ) Ignore this as it is periodic.
r r r r r b (=r b b b b b )
r r r r b b (=r r b b b b )
r r r b r b (=r b r b b b )
r r r b b b
r r b r r b (=b b r b b r ) Ignore this as it is periodic.
r r b r b b
r r b b r b

For a total of 6.
Jaap

Yes, I tried a few other small values of n using your method, and I do
indeed get the same numbers as in the OEIS sequence.

.

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 (or necklaces) with n red or blue beads ...
    (sci.math)
  • 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)
  • 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)
  • Re: Why are some constants not recognized where others of less importance are?
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    (sci.math)
  • 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, ...
    (sci.math)
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