A359865 a(n) is the number of k > 0 such that n-1-2*k >= 0 and a(n-1-2*k) * a(n-1) = a(n-1-k)^2.

0, 0, 0, 1, 1, 1, 1, 1, 2, 0, 0, 1, 2, 0, 2, 0, 3, 2, 3, 2, 1, 2, 2, 1, 0, 2, 3, 1, 0, 2, 2, 4, 4, 2, 3, 1, 4, 1, 1, 1, 4, 3, 1, 1, 5, 0, 2, 4, 4, 2, 1, 3, 0, 2, 0, 3, 1, 4, 2, 1, 5, 0, 3, 1, 5, 0, 4, 3, 0, 5, 1, 6, 1, 2, 3, 0, 6, 2, 4, 4, 2, 4, 3, 2, 2, 5, 2

Offset: 0

Author

Rémy Sigrist, Jan 16 2023

Keywords

• nonn

Comments

In other words, a(n) gives the number of geometric progressions (a(n-1-2*k), a(n-1-k), a(n-1)) of the form (x, x*y, x*y^2) or (x*y^2, x*y, x) with x, y >= 0.

This sequence has similarities with A308638: here we count geometric progressions, there arithmetic progressions.

Examples

			The first terms, alongside the corresponding k's, are:
  n   a(n)  k's
  --  ----  ------
   0     0  {}
   1     0  {}
   2     0  {}
   3     1  {1}
   4     1  {1}
   5     1  {2}
   6     1  {1}
   7     1  {1}
   8     2  {1, 2}
   9     0  {}
  10     0  {}
  11     1  {1}
  12     2  {1, 4}
  13     0  {}
  14     2  {3, 4}
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000
• Rémy Sigrist, Scatterplot of the first 250000 terms
• Rémy Sigrist, C program

Crossrefs

Cf. A308638.

Programs

C

See Links section.