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

0,5

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.

Links

Table of n, a(n) for n=0..68.

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

Formula

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

Example

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.

Prog

(C++) See Links section.

Crossrefs

Cf. A307097, A349991.

Sequence in context: A360853 A094848 A308740 * A117768 A018245 A264936

Adjacent sequences: A349987 A349988 A349989 * A349991 A349992 A349993

Keyword

nonn

Author

Rémy Sigrist, Dec 08 2021

Status

approved