A350226 a(n) is the length of the longest sequence of distinct numbers in arithmetic progression in the interval 0..n, ending with n and where the Thue-Morse sequence (A010060) is constant.

1, 1, 2, 2, 2, 2, 3, 3, 2, 4, 3, 2, 5, 3, 3, 6, 2, 3, 7, 4, 5, 3, 4, 3, 5, 5, 4, 4, 6, 3, 6, 7, 2, 4, 3, 5, 7, 7, 4, 6, 5, 5, 6, 3, 5, 6, 3, 6, 5, 7, 7, 5, 5, 4, 7, 4, 8, 6, 4, 4, 6, 6, 7, 8, 2, 3, 8, 5, 6, 3, 5, 5, 9, 7, 7, 7, 6, 4, 6, 7, 5, 5, 8, 5, 6, 6, 4

Offset

0,3

Comments

In other words, a(n) is the greatest k > 0 such that A010060(n) = A010060(n - i*d) for i = 0..k-1 and some d > 0 (see A350285 for the least such d).

This sequence is unbounded (this is a consequence of Van der Waerden's theorem).

Links

Rémy Sigrist, Table of n, a(n) for n = 0..8191

Rémy Sigrist, C program for A350226

Example

For n = 12:
- the first 13 terms of A010060 are:
         0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0
         ^        ^        ^        ^        ^
- A010060(0) = A010060(3) = A010060(6) = A010060(9) = A010060(12),
- and there is no longer sequence of distinct numbers <= 12 in arithmetic progression ending in 12 with this property,
- so a(12) = 5.

Prog

(C) See Links section.

Crossrefs

Cf. A005346, A010060, A342818, A342827, A350235 (records), A350285 (least first differences).

Sequence in context: A143901 A115263 A055894 * A362136 A224713 A357427

Adjacent sequences: A350223 A350224 A350225 * A350227 A350228 A350229

Keyword

nonn

Author

Rémy Sigrist, Dec 20 2021

Status

approved