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A185436 The minimum number of colors required to color an n-bead bracelet so that each bead can be uniquely identified by its color and the color(s) of its two immediately-adjacent neighbors. 1
1, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4 (list; graph; refs; listen; history; text; internal format)



I am not sure whether or not this sequence is just the biggest number k for which A002411(k) is less than or equal to n. Clearly that forms a lower bound, since it is the number of 3-symbol strings where reversing the string doesn't matter.


Table of n, a(n) for n=1..19.

Canadian Mathematics Olympiad, Problem 2, 1984.

MathLearning.org, Solution [gives A185437]


For example, the string AABABBBCCC colors a bracelet of 10 beads using 3 symbols. No two beads have the same color and neighbors with the same set of colors. In this problem, the order in which the neighbors' colors occur doesn't matter (because the bracelet can be turned over). So the string ABBABCBCAC wouldn't work, because we can't distinguish between the second and third beads (ABB and BBA). We can't do this with only 2 colors, so a(10) = 3.


Cf. A185437, A002411

Sequence in context: A065682 A065681 A121242 * A025794 A025793 A102680

Adjacent sequences:  A185433 A185434 A185435 * A185437 A185438 A185439




Jack W Grahl, Jan 27 2011


Definition and example changed (per a comment by Wouter Meeussen) to refer to the ring of beads using the term "bracelet," rather than "necklace," since turning the ring over is allowed. - Jon E. Schoenfield, Sep 18 2013



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Last modified January 25 11:11 EST 2021. Contains 340416 sequences. (Running on oeis4.)