A249872 Number of iterations to reach a final state for an n X n lattice of sandpiles on a torus according to rules specified in the comment section.

0, 7, 28, 133, 316, 913, 1360, 2987, 4340, 7495, 11328, 17166, 23032, 32903, 42440, 61146, 72872, 98243, 119232, 153173, 175356, 231023, 271828, 333160, 397736, 474983, 543904, 647743, 744408, 873435, 965556, 1142970, 1270772, 1489867, 1655876, 1901359

Offset

1,2

Comments

Let the lattice be c[i,j], 0 <= i,j < n. Fill each cell except c[0,0] with 4 grains of sand. Until all c[i,j] < 4, do the following:

Find a c[i,j] >= 4. (According to Knuth, it does not matter which cell is chosen, the result will be the same.) Decrement the chosen cell by 4 and increment its 4 neighbors by 1. c[0,0] is never increased, sand grains placed here are lost. The number of iterations needed is a(n).

Links

Joerg Arndt, Table of n, a(n) for n = 1..600 (terms for n<=200 from Lars Blomberg)

Donald E. Knuth, Sand Piles and Spanning Trees, Computer Musings 2004.

Example

For n=3 the iterations start
   4* 4       0  5*       1  1       1  1       2  1
4  4  4 -> 4  5  4 ->  4* 5  5 -> 0  6* 6 -> 1  2  7*  ...
4  4  4    4  5  4     4  5  5    5  5  5    5  6  5
and end
        2  1        2  2       2  2       3  2        3  3
...  1  0  5* -> 2  1  1 -> 3  1  1 -> 3  2  1 ->  3  2  2
     5  4  3     5* 4  4    1  5* 5    2  1  6*    3  2  2
where * indicates the cell being processed.

Crossrefs

Sequence in context: A316106 A359203 A354456 * A238448 A290356 A025030

Adjacent sequences: A249869 A249870 A249871 * A249873 A249874 A249875

Keyword

nonn

Author

Lars Blomberg, Nov 07 2014

Status

approved