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%I #17 Dec 09 2017 19:45:05
%S 1,2,3,3,3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
%T 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
%U 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2
%N The least number of colors required to color an n-bead necklace so that each bead can be identified.
%C In other words, the least number of colors in any coloring of the necklace that is not symmetric under any element of the corresponding dihedral group.
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%F a(n) = 2 for all n > 5.
%F G.f.: x*(x^2+1)*(x^3-x-1)/(x-1). [_Colin Barker_, Oct 26 2012]
%e For n=5, one coloring is ABBCC. Any coloring using two symbols will have two indistinguishable beads.
%e For n > 5, a coloring is ABAAB...B, where ... is zero or more B's. We can tell the A's apart because one has a B on either side, of the other two one is closer to the single B, and one is closer to the long sequence of B's. Of the B's, one has an A on either side. The remaining B's can be distinguished by counting along the string of B's starting at the end with a singleton A.
%Y Cf. A185436.
%K nonn,easy
%O 1,2
%A _Jack W Grahl_, Jan 27 2011
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