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A185437 The least number of colors required to color an n-bead necklace so that each bead can be identified. 1


%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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Last modified December 8 14:38 EST 2019. Contains 329865 sequences. (Running on oeis4.)