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A351379
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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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3
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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
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OFFSET
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2,1
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COMMENTS
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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.
See A307652 for details of the sandpile group identity.
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LINKS
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Luis David Garcia-Puente and Brady Haran, Sandpiles, Numberphile video, YouTube.com, Jan. 13, 2017.
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FORMULA
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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.
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EXAMPLE
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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.
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CROSSREFS
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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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