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A307652
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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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5
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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
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OFFSET
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1,1
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COMMENTS
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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).
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LINKS
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Luis David Garcia-Puente and Brady Haran, Sandpiles, Numberphile video, YouTube.com, Jan. 13, 2017.
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.
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FORMULA
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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.
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EXAMPLE
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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.
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CROSSREFS
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Cf. A007341 (order of the sandpile group of the (n-1)X(n-1) grid graph).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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