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.

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

0,2

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.

Links

Table of n, a(n) for n=0..40.

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.

Crossrefs

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

Sequence in context: A228458 A203050 A203332 * A190065 A293068 A213294

Adjacent sequences: A351780 A351781 A351782 * A351784 A351785 A351786

Keyword

nonn

Author

Scott R. Shannon and Zach J. Shannon, Feb 19 2022

Status

approved