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%I #32 Jul 17 2021 07:00:10
%S 1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,
%T 4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,
%U 5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6
%N 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.
%C 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.
%H Canadian Mathematics Olympiad, <a href="http://www.math.ca/Competitions/CMO/examt/english84.pdf">Problem 2</a>, 1984.
%H MathLearning.org, <a href="http://schoolexercisebooks.blogspot.com/2010/09/16th-canadian-mathematical-olympiad.html">Solution</a> [gives A185437]
%e 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.
%Y Cf. A185437, A002411
%K nonn
%O 1,2
%A _Jack W Grahl_, Jan 27 2011
%E 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
%E More terms from _Bert Dobbelaere_, Jul 17 2021
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