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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.
3
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
KEYWORD
nonn
AUTHOR
Lars Blomberg, Nov 07 2014
STATUS
approved