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A249872 Number of iterations to reach a final state for an n X n lattice of sand piles on a torus according to rules specified in the comment section. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A054626 A220361 A219737 * A238448 A290356 A025030

Adjacent sequences:  A249869 A249870 A249871 * A249873 A249874 A249875

KEYWORD

nonn

AUTHOR

Lars Blomberg, Nov 07 2014

STATUS

approved

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Last modified November 18 17:56 EST 2017. Contains 294894 sequences.