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

Author

Lars Blomberg, Nov 07 2014

Keywords

• nonn

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).

Examples

			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.
		

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.