A359866 a(n) is the number of k > 0 such that n-1-2*k >= 0 and a(n-1-2*k) >= a(n-1-k) >= a(n-1).

0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 3, 0, 4, 0, 3, 0, 3, 1, 0, 7, 0, 6, 0, 7, 0, 6, 0, 7, 1, 0, 9, 0, 10, 0, 8, 0, 10, 0, 9, 0, 10, 1, 0, 11, 0, 14, 0, 13, 0, 14, 0, 12, 0, 13, 1, 1, 3, 0, 17, 0, 17, 0, 18, 0, 17, 0, 18, 0, 17, 1, 2, 3, 0, 21, 0, 23, 0, 22, 0, 23, 0

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

Cf. A359867.

Sequence in context: A272474 A340426 A308717 * A297217 A218859 A363028

Adjacent sequences: A359863 A359864 A359865 * A359867 A359868 A359869

Keyword

nonn,look

Author

Rémy Sigrist, Jan 16 2023

Status

approved