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, 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, 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, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset

1,2

Comments

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.

Links

Table of n, a(n) for n=1..90.

Canadian Mathematics Olympiad, Problem 2, 1984.

MathLearning.org, Solution [gives A185437]

Example

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.

Crossrefs

Cf. A185437, A002411

Sequence in context: A065682 A065681 A121242 * A025794 A354760 A025793

Adjacent sequences: A185433 A185434 A185435 * A185437 A185438 A185439

Keyword

nonn

Author

Jack W Grahl, Jan 27 2011

Extensions

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

More terms from Bert Dobbelaere, Jul 17 2021

Status

approved