A225179 a(n) = min{2 + c_n + 2(c_1 + c_2 + ... + c_(n-1)) | n = p + q, 1 <= p < q < n, gcd(p, q) = 1, and p/q has continued fraction expansion [0; c_1, c_2, ...., c_n]}.

1, 3, 4, 5, 6, 7, 7, 8, 8, 9, 9, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 14, 15, 15, 15, 15, 16, 15, 15, 15, 16, 15, 16, 16, 16, 16, 16, 16, 17, 16, 17, 16, 17, 17, 16, 17, 17, 17, 17, 17, 18, 17, 17, 17, 17, 17, 19, 17, 18, 18, 17, 18, 18, 18, 18, 18, 19, 18

Offset

1,2

Comments

It is conjectured that a(n) = min{ B(w) | w is unbordered and length(w) = n }, where B(w) is the number of distinct unbordered factors in w.

If this conjecture were to be proved, we could use this is the definition, and put the present definition into the FORMULA field. It is known to be true for n <= 20.

[A word v is bordered if there is a nonempty word u different from v which is both a prefix and a suffix of v (see, for example, Allouche and Shallit).]

References

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 28.

Kalle Saari, Open problem presented at Workshop on Challenges in Combinatorics on Words, Fields Institute, Toronto, April 21-26, 2013.

Links

Kalle Saari, Table of n, a(n) for n = 1..199

Kalle Saari, Unbordered words with the smallest number of distinct unbordered factors

Crossrefs

Sequence in context: A095254 A262980 A242374 * A121857 A121854 A196119

Adjacent sequences: A225176 A225177 A225178 * A225180 A225181 A225182

Keyword

nonn

Author

N. J. A. Sloane, May 03 2013, based on an email from Kalle Saari, Apr 23 2013

Status

approved