A349990 For any n >= 0, consider a sandpile model on the infinite square lattice starting with n grains at the origin, the other sites being empty; a(n) gives the number of nonempty sites after stabilization of this sandpile model.

0, 1, 1, 1, 4, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 12, 13, 13, 13, 12, 13, 13, 13, 12, 13, 13, 13, 16, 17, 17, 17, 20, 21, 21, 21, 20, 21, 21, 21, 20, 21, 21, 21, 24, 25, 25, 25, 24, 25, 25, 25, 24, 25, 25, 25, 32, 33, 33, 33, 36, 37, 37, 37, 36, 37, 37, 37, 36

Offset: 0

Author

Rémy Sigrist, Dec 08 2021

Keywords

• nonn

Comments

A site is unstable when it holds 4 or more grains.

As long as there is an unstable site:

- choose such an unstable site,

- remove 4 grains from this site and add 1 grain to each of its four neighbors.

This procedure is guaranteed to result in a stable configuration, which does not depend on the order in which we treat the unstable sites.

Examples

			For n = 25:
- after stabilization, we have the following configuration:
         1
       2 3 2
     1 3 1 3 1
       2 3 2
         1
- there are 13 nonempty sites,
- so a(25) = 13.
		

Links

• Rémy Sigrist, Colored representation of the stabilized configuration for n = 1000000 (white, green, purple and gold pixels correspond to sites with 0, 1, 2 and 3 grains, respectively)
• Rémy Sigrist, C++ program for A349990
• Wikipedia, Sandpile models on infinite grids

Crossrefs

Cf. A307097, A349991.

Formula

a(4*n) + 1 = a(4*n+1) = a(4*n+2) = a(4*n+3).