OFFSET
0,11
COMMENTS
This sequence is unbounded:
- by contradiction: suppose that a(n) < M for some fixed M,
- then, by Van der Waerden's theorem, we have an arithmetic progression of 2*M+1 indices where the sequence has the same value: say a(m) = a(m + k*r) for k = 0..2*M with m >= 0 and r > 0,
- this would imply that a(m + 2*M*r + 1) >= M, a contradiction.
This sequence has infinitely many 0's (if a(m) < a(n) for any m < n, then a(n+1) = 0).
LINKS
Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Scatterplot of the first 250000 terms
Rémy Sigrist, C program
Wikipedia, Van der Waerden's theorem.
EXAMPLE
The first terms, alongside the corresponding k's, are:
n a(n) k's
-- ---- ------------
0 0 {}
1 0 {}
2 0 {}
3 1 {1}
4 0 {}
5 1 {2}
6 0 {}
7 1 {2}
8 1 {2}
9 0 {}
10 3 {1, 2, 3}
11 0 {}
12 4 {2, 3, 4, 5}
PROG
(C) See Links section.
CROSSREFS
KEYWORD
nonn,look
AUTHOR
Rémy Sigrist, Jan 16 2023
STATUS
approved