A349990 -id:A349990 - cp's OEIS frontend

A349991 For any n >= 0, consider a sandpile model on the infinite hexagonal 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, 1, 1, 6, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 18, 19, 19, 19, 19, 19, 18, 19, 19, 19, 19, 19, 18, 19, 19, 19, 19, 19, 18, 19, 19, 19, 19, 19, 18, 19, 19, 19, 19, 19, 18, 19, 19, 19, 19, 19, 24, 25

Offset: 0

Rémy Sigrist, Dec 08 2021

nonn

A site is unstable when it holds 6 or more grains.
As long as there is an unstable site:
- choose such an unstable site,
- remove 6 grains from this site and add 1 grain to each of its six 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.

			For n = 54:
- after stabilization, we have the following configuration:
             2
          4     4
       2     3     2
          3     3
       4           4
          3     3
       2     3     2
          4     4
             2
- we have 18 nonnempty sites,
- so a(54) = 18.

Rémy Sigrist, Colored representation of the stabilized configuration for n = 1000000 (white, green, purple, gold, blue and red pixels correspond to sites with 0, 1, 2, 3, 4 and 5 grains, respectively)
Rémy Sigrist, C++ program for A349991
Wikipedia, Sandpile models on infinite grids

Cf. A349990.

a(6*n) + 1 = a(6*n + k) for k = 1..5.

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.

Original entry on oeis.org

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

Rémy Sigrist, Mar 09 2022

nonn

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.

			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.

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)

Cf. A349990, A351783, A352226.

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]))) }

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

A351784 Number of cells containing one or more grains of sand after n grains of sand are added to one cell in an initially empty and infinite 3D cubic grid for the 3D sandpile model.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 6, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 24, 25, 25, 25, 25, 25, 24, 25, 25, 25, 25, 25, 24, 25, 25, 25, 25, 25, 24, 25, 25, 25, 25, 25, 24, 25, 25, 25, 25, 25, 24, 25, 25, 25, 25, 25, 24, 25, 25, 25, 25, 25, 24, 25, 25

Offset: 0

Scott R. Shannon and Zach J. Shannon, Feb 19 2022

nonn

The 3D sandpile model follows the same rules as the 2D model except that cells topple and transfer one grain of sand to their six nearest neighbors when the cell contains 6 or more grains. Cells containing 0 to 5 grains are stable.

Scott R. Shannon, Table of n, a(n) for n = 0..10000
Per Bak, Chao Tang, and Kurt Wiesenfeld, Self-organized criticality: An explanation of the 1/f noise, Phys. Rev. Lett. 59 (1987), 381-384.
Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter, and Tianyuan Xu, Sandpiles and Dominos, Electronic Journal of Combinatorics, Volume 22(1), 2015.
Luis David Garcia-Puente and Brady Haran, Sandpiles, Numberphile video, YouTube.com, Jan. 13, 2017.
Zach J. Shannon, Image of the occupied cells for a(96)=44, bisected along the y-z plane. The colors are red=1 (24 total cells), blue=3 (14 total cells), violet=5 (6 total cells) grains per cell.

Cf. A351783, A351379, A349990 (2D version), A307652, A259013, A180230.

A352226 Consider a 2D sandpile model where each site with 2 or more grains, say at location (x, y), topples and transfers one grain of sand to the sites at locations (x+1, y) and (x, y+1). Let S(n) be the configuration after stabilization of a configuration with n grains at the origin. a(n) = Max_{ (x,y) in S(n) } (x+y).

Original entry on oeis.org

0, 1, 1, 3, 3, 3, 3, 5, 5, 5, 5, 7, 7, 7, 7, 9, 9, 9, 9, 9, 9, 9, 9, 11, 11, 11, 11, 13, 13, 13, 13, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 21, 21, 21, 21, 23, 23, 23, 23, 23, 23, 23, 23

Offset: 1

Rémy Sigrist, Mar 08 2022

nonn

Sites containing 0 or 1 grain are stable. S(n) contains n elements.

			For n = 15:
- S(15) corresponds to the following configuration:
    4|  X X X
    3|X   X   X
    2|X     X X
    1|X       X
    0|X X X X
     +---------
      0 1 2 3 4
- x+y is maximized for (x,y) = (4,3) and (3,4),
- so a(15) = 3+4 = 7.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Representation of the configuration for n = 100000
Wikipedia, Sandpile models on directed graphs

Cf. A180230, A349990, A350188, A351783.

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

cp's OEIS Frontend

A349991 For any n >= 0, consider a sandpile model on the infinite hexagonal 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.

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A351784 Number of cells containing one or more grains of sand after n grains of sand are added to one cell in an initially empty and infinite 3D cubic grid for the 3D sandpile model.

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