A circular necklace contains n beads. Each bead is black or white. How many different necklaces can be made with n beads?

There is no clasp to identify a specific point on the chain, and a flipped over necklace is still the same necklace.
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To get you started:

With 1 bead, the necklace can be either 1 black or 1 white bead.

With 2 beads, the necklace can be either 2 black, 2 white, or 1 black-1 white

With 3 beads, the necklace can be either 3 black, 3 white, 2 black-1 white, 2 white-1 black, etc...

# Beads  Number of Necklaces
   1          2
   2          3
   3          4
   4          6
   5          8
   6         13

ID Number: A000029 (Formerly M0563 and N0202)
URL: http://www.research.att.com/projects/OEIS?Anum=A000029
Sequence: 1,2,3,4,6,8,13,18,30,46,78,126,224,380,687,1224,2250,4112,
7685,14310,27012,50964,96909,184410,352698,675188,1296858,
2493726,4806078,9272780,17920860,34669602,67159050,
130216124,252745368,490984488
Name: Number of necklaces with n beads of 2 colors, allowing turning over.
References N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958),
285-302.
.............AND OTHERS

ady

Posted by Ady TZIDON on 2004-08-19 03:27:45