A350188 Consider a 2D sandpile model where each site with 3 or more grains, say at location (x, y), topples and transfers one grain of sand to the sites at locations (x+1, y-1), (x+1, y) and (x+1, y+1); a(n) gives the number of nonempty sites after stabilization of a configuration starting with n grains at the origin.

0, 1, 1, 3, 4, 4, 3, 4, 4, 7, 8, 8, 10, 11, 11, 10, 11, 11, 14, 15, 15, 17, 18, 18, 17, 18, 18, 21, 22, 22, 24, 25, 25, 24, 25, 25, 27, 28, 28, 30, 31, 31, 30, 31, 31, 36, 37, 37, 39, 40, 40, 39, 40, 40, 39, 40, 40, 42, 43, 43, 42, 43, 43, 45, 46, 46, 48, 49

Offset

0,4

Comments

Sites containing 0, 1 or 2 grains are stable.

After stabilization, there are:

- 2*a(n) - n sites with one grain,

- n - a(n) sites with two grains.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Rémy Sigrist, Representation of the configuration for n = 100000 (blue pixels correspond to sites with one grain, red pixels to sites with two grains)

Formula

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

Example

For n = 10 :
- the model evolves (for example) as follows:
                       1          1
            3        . 2        . 2 1
  10  ->  1 3  ->  1 . 3  ->  1 . . 1
            3        . 2        . 2 1
                       1          1
- there are 8 nonempty sites in the stabilized configuration,
- so a(10) = 8.

Prog

(PARI) a(n) = { my (s=[n], v=0); for (k=-1, oo, if (vecmax(s)==0, return (v), v += sum(k=1, #s, s[k]%3>0); s \= 3; s = concat([ s , [0], [0]]) + concat([[0],  s , [0]]) + concat([[0], [0],  s ]); while (#s>2 && s[1]==0, s = s[2..#s-1]))) }

Crossrefs

Cf. A349990, A351783, A352226.

Sequence in context: A306477 A276865 A253718 * A374363 A295570 A035004

Adjacent sequences: A350185 A350186 A350187 * A350189 A350190 A350191

Keyword

nonn

Author

Rémy Sigrist, Mar 09 2022

Status

approved