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
KEYWORD
nonn
AUTHOR
Lars Blomberg, Nov 07 2014
STATUS
approved