A307097 Number of configurations in the repeating cycle of the sandpile model in a bounded square of size 2n+1.

2, 4, 4, 4, 6, 6, 8, 6, 10, 18, 26, 10, 10, 8, 12, 30, 8, 32, 14, 32, 52, 26, 60, 52, 34, 74, 14, 80, 36, 38, 24, 54, 26, 30, 36, 58, 22, 14, 26, 36, 38, 20, 36, 60, 24, 24, 18, 14, 24, 34, 70, 104, 48, 56, 36, 50, 50, 48, 152, 28, 110, 30, 172, 64, 104, 158, 150, 60, 36, 186, 52, 50

Offset: 1

Author

Scott R. Shannon, Mar 24 2019

Keywords

• nonn

Comments

The Abelian sandpile model considers the behavior of grains of sand on a square grid when a square topples sand to its nearest neighbors when the number of grains in the square is greater than or equal to 4, where 4 is the number of nearest neighbors. If the grid is instead bounded by a square box then it is natural to extend this rule so that squares on the border also topple when they contain more or the same number of grains as the number of their nearest neighbors, i.e., 2 for corner squares, 3 for edge squares. Unlike the standard Abelian sandpile model on a finite grid, the square grid in this model does not lose sand, and assuming one keeps adding sand to the grid after each topple stabilization, eventually a critical number of grains will be added such that the resulting configurations will cycle. This sequence {a(n)} is the number of configurations in the cycle, for the sandpile model bounded by a square of size 2n+1, assuming the square grid starts with no sand and sand is continuously added to the center square until a cycle first occurs. As the toppling in a given area will repeat indefinitely, once the repeating cycle state is reached, one cannot topple the unstable squares in any random order and reach a stable configuration. To avoid such issues all squares are toppled simultaneously.

Examples

			For n=1 the square size is 2*1+1 = 3. The number of sand grains in each square is shown below, after the addition of 4, 8 and 12 grains to the central square:
.
   After 4         After 8         After 12
+---+---+---+   +---+---+---+   +---+---+---+
| 0 | 1 | 0 |   | 0 | 2 | 0 |   | 0 | 3 | 0 |
+---+---+---+   +---+---+---+   +---+---+---+
| 1 | 0 | 1 |   | 2 | 0 | 2 |   | 3 | 0 | 3 |
+---+---+---+   +---+---+---+   +---+---+---+
| 0 | 1 | 0 |   | 0 | 2 | 0 |   | 0 | 3 | 0 |
+---+---+---+   +---+---+---+   +---+---+---+
.
The edge squares now contain 3 grains, which means they are unstable as 3 equals their nearest neighbor count. This configuration thus topples to:
.
+---+---+---+
| 2 | 0 | 2 |
+---+---+---+
| 0 | 4 | 0 |
+---+---+---+
| 2 | 0 | 2 |
+---+---+---+
.
The central square is again unstable, as are the four corner squares as they contain 2 grains. This configuration topples to:
.
+---+---+---+
| 0 | 3 | 0 |
+---+---+---+
| 3 | 0 | 3 |
+---+---+---+
| 0 | 3 | 0 |
+---+---+---+
.
This is the same as the configuration after 12 grains above. These last two configurations cycle forever, thus a(1) = 2.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..500
• Per Bak, Chao Tang, 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.
• L. Levine, W. Pegden, C. K. Smart, Apollonian structure in the Abelian sandpile, Geom. Funct. Anal. (2016) 26:306.
• Scott R. Shannon, Cycle for square of size 2*2+1 = 5. This cycle of 4 configurations is reached after the addition of 40 grains of sand. This is the first square size in a sequence of three sizes that have a cycle of 4: a(2), a(3), a(4). For this, and other images, white=0, yellow=1, green=2, blue=3, red=4, black>=5 grains.
• Scott R. Shannon, Cycle for square of size 2*3+1 = 7. The second square to have a cycle of 4. This is reached after the addition of 100 grains.
• Scott R. Shannon, Cycle for square of size 2*4+1 = 9. The third square to have a cycle of 4. This is reached after the addition of 160 grains.
• Scott R. Shannon, Complete growth and cycle for square of size 2*7+1 = 15. This shows the addition of 468 grains (each image is pre-topple after the central square has regained 4 grains) followed by each iteration for the resulting 8-cycle configuration. The cycle is shown ten times.
• Wikipedia, Abelian Sandpile Model.

Crossrefs

Cf. A259013, A180230, A056219, A307652.

Cf. A007341 (order of the sandpile group of the (n-1)X(n-1) grid graph).