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.

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

Author

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

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• 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.

Crossrefs

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

Formula

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.