A259013 -id:A259013 - cp's OEIS frontend

A307652 The number of grains of sand in the identity element for the sandpile group on an (n+1) X (n+1) square grid.

8, 12, 40, 52, 72, 88, 136, 160, 216, 244, 320, 356, 408, 448, 544, 592, 704, 756, 888, 948, 1088, 1156, 1304, 1376, 1504, 1584, 1736, 1820, 1984, 2076, 2288, 2384, 2536, 2640, 2912, 3024, 3200, 3316, 3624, 3748, 3976, 4104, 4392, 4528, 4824, 4968, 5216, 5364, 5664, 5820, 6088, 6248, 6616

Offset: 1

Scott R. Shannon, Apr 20 2019

nonn

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. Squares on the edge of the board lose sand from the grid when toppling thus a stable configuration for the grid will always occur after a finite number of topples. Starting with the maximal stable grid consisting of 3 grains of sand in all squares, adding sand to one or more squares, and then performing topple stabilization results in a set of recurrent configurations which form the elements of the sandpile group for the given grid size. This group includes one configuration which acts as the identity element for the group, i.e., adding the identity configuration to any chosen group element and then performing topple stabilization results in the chosen group element.

This sequence {a(n)} is the number of sand grains in the identity element of the sandpile group on a square grid of size (n+1) X (n+1).

			a(1) = 2 X 2 grid.
       Identity: | 2 2 |
                 | 2 2 | = 8 grains.
a(2) = 3 X 3 grid.
       Identity: | 2 1 2 |
                 | 1 0 1 |
                 | 2 1 2 | = 12 grains.
a(3) = 4 X 4 grid.
       Identity: | 2 3 3 2 |
                 | 3 2 2 3 |
                 | 3 2 2 3 |
                 | 2 3 3 2 | = 40 grains.
a(4) = 5 X 5 grid.
       Identity: | 2 3 2 3 2 |
                 | 3 2 1 2 3 |
                 | 2 1 0 1 2 |
                 | 3 2 1 2 3 |
                 | 2 3 2 3 2 | = 52 grains.

Scott R. Shannon, Table of n, a(n) for n = 1..500
Yvan Le Borgne, Dominique Rossin, On the identity of the sandpile group. Discrete Mathematics, 256 (2002) 775-790.
Luis David Garcia-Puente and Brady Haran, Sandpiles, Numberphile video, YouTube.com, Jan. 13, 2017.
Alexander E. Holroyd et al., Chip-firing and Rotor-Routing on Directed Graphs. arXiv:0801.3306v4 [math.CO], 2013.
Scott R. Shannon, Identity for the 50x50 grid. For this, and other images, black=0, yellow=1, blue=2, red=3 grains.
Scott R. Shannon, Identity for the 51x51 grid. This shows the crossed line pattern through the center of the grid which is typical of grids with odd numbered side lengths.
Scott R. Shannon, Identity for the 1000x1000 grid.
Scott R. Shannon, Identity for the 4000x4000 grid. This contains 37246680 grains.
Scott R. Shannon, Simplified Java code for finding the identity element and the sequence a(n).
Wikipedia, Abelian Sandpile Model.

Cf. A259013, A180230, A300006.

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

Identity element = ([6n] - ([6n])*)* , where [6n] is the all 6's grid of size (n+1) X (n+1), and (x)* represents the topple stabilization of the grid x.

The sequence is closely fitted by the quadratic a(n) ~ 2.32*n^2, where 2.32 corresponds to the approximate grains per square density of the identity element configurations.

A351783 Number of grains of sand required to be added to one cell at the origin in an initially empty and infinite 3D cubic grid for the 3D sandpile model such that the distance from the origin of the furthest nonempty cell along the axes is n.

Original entry on oeis.org

0, 6, 36, 162, 516, 1230, 2430, 3756, 5862, 9036, 12822, 16710, 22182, 28758, 36144, 45444, 54966, 66270, 78870, 93834, 109866, 127260, 146412, 169698, 192366, 218214, 244752, 273480, 307224, 341430, 380988, 420558, 463350, 510024, 558090, 611088, 664494, 723060, 784014, 844134, 921486

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.

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(40)=921486. For this and the below image, red=1, green=2, blue=3, violet=4, orange=5 grains per cell. The axes are in the middle of the red squares.
Zach J. Shannon, Image of the occupied cells for a(40)=921486, bisected along the y-z plane.

Cf. A351784, A351379, A307652, A259013, A180230.

A351379 The number of grains of sand in the identity element for the 3D sandpile group on an n X n X n cubic grid.

Original entry on oeis.org

24, 54, 288, 480, 744, 1062, 1968, 2616, 3480, 4398, 6000, 7344, 9744, 11628, 14256, 16632, 20376, 23436, 27312, 30984, 37104, 41652, 47424, 52776, 60432, 66636, 75552, 82752, 93288, 101676, 112488, 121968, 135768, 146436, 163032, 175182, 191256, 204690, 221784, 236646, 257400, 273738, 296784

Offset: 2

Scott R. Shannon and Zach J. Shannon, Feb 09 2022

nonn

See A307652 for details of the sandpile group identity.

			a(2) = 2 X 2 X 2 grid. Identity:
       Layer 1: | 3 3 |  Layer 2: | 3 3 |
                | 3 3 |           | 3 3 |  = 24 grains.
a(3) = 3 X 3 X 3 grid. Identity:
       Layer 1: | 3 2 3 |  Layer 2: | 2 1 2 |  Layer 3: | 3 2 3 |
                | 2 1 2 |           | 1 0 1 |           | 2 1 2 |
                | 3 2 3 |           | 2 1 2 |           | 3 2 3 |  = 54 grains.

Noah Doman, The Identity of the Abelian Sandpile Group, Bachelor Thesis, University of Groningen, January 2020.
Luis David Garcia-Puente and Brady Haran, Sandpiles, Numberphile video, YouTube.com, Jan. 13, 2017.
Yvan Le Borgne and Dominique Rossin, On the identity of the sandpile group, Discrete Mathematics, 256 (2002) 775-790.
Scott R. Shannon, Middle layer of the 100x100x100 identity. This contains 3486864 grains. For this and other images, white=0, red=1, green=2, blue=3, violet=4, yellow=5 grains per cell.
Scott R. Shannon, Top layer of the 100x100x100 identity.
Scott R. Shannon, Middle layer of the 101x101x101 identity. Similarly to the 2D sandpile model, when n is odd the middle layers have a cross-like pattern.
Zach J. Shannon, 3D image of the full 80x80x80 identity. The same colors as above are used except cells with no grains are shown as vacancies, not white.
Zach J. Shannon, 3D image of half the 80x80x80 identity.
Zach J. Shannon, 3D image of half the 80x80x80 identity without the cells containing 5 grains.
Zach J. Shannon, 3D image of half the 80x80x80 identity showing only the cells containing 5 grains.

Cf. A307652 (square grid), A259013, A180230, A300006, A007341.

Identity element = ([10n] - ([10n])*)* , where [10n] is the all 10's grid of size n X n X n, and (x)* represents the topple stabilization of the grid x.

The sequence is approximately fitted by the cubic a(n) ~ 3.48*n^3, where 3.48 corresponds to the approximate grains-per-cube density of the identity element configurations.

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

Original entry on oeis.org

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

Scott R. Shannon, Mar 24 2019

nonn

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.

			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.

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.

Cf. A259013, A180230, A056219, A307652.

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

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

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.

A328506 Iteration of Abelian sandpile model where the n-th matrix expansions occurs. Begins with infinite sand in 1 X 1 matrix.

Original entry on oeis.org

1, 5, 16, 36, 66, 101, 160, 218, 285, 374, 464, 565, 680, 815, 969, 1124, 1282, 1467, 1659, 1863, 2091, 2346, 2559, 2824, 3100, 3411, 3690, 4043, 4380, 4697, 5060, 5468, 5833, 6266, 6670, 7132, 7595, 8006, 8502, 9004, 9518, 10039, 10609, 11155, 11740, 12304, 12971, 13603, 14202, 14861, 15532, 16217

Offset: 1

Parker Grootenhuis, Oct 22 2019

nonn

The Abelian sandpile model is a cellular automaton considering the behavior of sand grains on a square grid. Any square containing 4 or more grains will topple, sending a grain to each of its 4 neighbors and subtracting 4 grains from itself.

Here, expansion refers to the addition of a boundary layer to the outside of the existing matrix when the model reaches beyond the previous matrix boundary.

			                                                      _ _ _ _ _
          _ _ _      _ _ _      _ _ _      _ _ _     |0|0|1|0|0|
   _     |0|1|0|    |0|2|0|    |0|3|0|    |0|4|0|    |0|2|1|2|0|
  |∞| -> |1|∞|1| -> |2|∞|2| -> |3|∞|3| -> |4|∞|4| -> |1|1|∞|1|1| -> ...
   ‾     |0|1|0|    |0|2|0|    |0|3|0|    |0|4|0|    |0|2|1|2|0|
          ‾ ‾ ‾      ‾ ‾ ‾      ‾ ‾ ‾      ‾ ‾ ‾     |0|0|1|0|0|
                                                      ‾ ‾ ‾ ‾ ‾
            ^                                             ^
     1st expansion on                              2nd expansion on
   1st iteration (a(1) = 1)                      5th iteration (a(2) = 5)

Cf. A259013, A249872.

L = 3;
plane = zeros(3,3);
plane(2,2) = 99999999999999999999999999999999999999999999999;
listn = [];
for n = 1:50000
    plane2 = plane;
    for r = 1:L
        for c = 1:L
            if plane(r,c) > 3
                plane2(r,c) = plane2(r,c) - 4;
                plane2(r-1,c) = plane2(r-1,c)+1;
                plane2(r+1,c) = plane2(r+1,c)+1;
                plane2(r,c-1) = plane2(r,c-1)+1;
                plane2(r,c+1) = plane2(r,c+1)+1;
            end
        end
    end
    if sum(plane2(:,1))+sum(plane2(1,:)) > 0
        plane2 = padarray(plane2,[1,1]);
        L = L+2;
        listn = [listn n];
    end
    plane = plane2;
end
fprintf('%s\n', sprintf('%d,', listn))

Step(M)={my(n=#M, R=matrix(n,n)); for(i=2, n-1, for(j=2, n-1, if(M[i,j]>=4, R[i,j]-=4; R[i,j+1]++; R[i,j-1]++; R[i-1,j]++; R[i+1,j]++))); M+R}
Expand(M)={my(n=#M, R=matrix(n+2, n+2)); for(i=1, n, for(j=1, n, R[i+1, j+1]=M[i,j])); R}
seq(n)={my(L=List(), M=matrix(3,3), k=0); while(#LAndrew Howroyd, Oct 23 2019

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A307652 The number of grains of sand in the identity element for the sandpile group on an (n+1) X (n+1) square grid.

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A351783 Number of grains of sand required to be added to one cell at the origin in an initially empty and infinite 3D cubic grid for the 3D sandpile model such that the distance from the origin of the furthest nonempty cell along the axes is n.

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A351379 The number of grains of sand in the identity element for the 3D sandpile group on an n X n X n cubic grid.

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

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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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A328506 Iteration of Abelian sandpile model where the n-th matrix expansions occurs. Begins with infinite sand in 1 X 1 matrix.

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