|
%I #29 Mar 11 2022 08:03:48
%S 0,7,28,133,316,913,1360,2987,4340,7495,11328,17166,23032,32903,42440,
%T 61146,72872,98243,119232,153173,175356,231023,271828,333160,397736,
%U 474983,543904,647743,744408,873435,965556,1142970,1270772,1489867,1655876,1901359
%N 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.
%C 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:
%C 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).
%H Joerg Arndt, <a href="/A249872/b249872.txt">Table of n, a(n) for n = 1..600</a> (terms for n<=200 from Lars Blomberg)
%H Donald E. Knuth, <a href="https://www.youtube.com/watch?v=h25QHIE4d2s&list=PLoROMvodv4rNMsVRnSJ44WuwbminUqXX2&index=17">Sand Piles and Spanning Trees</a>, Computer Musings 2004.
%e For n=3 the iterations start
%e 4* 4 0 5* 1 1 1 1 2 1
%e 4 4 4 -> 4 5 4 -> 4* 5 5 -> 0 6* 6 -> 1 2 7* ...
%e 4 4 4 4 5 4 4 5 5 5 5 5 5 6 5
%e and end
%e 2 1 2 2 2 2 3 2 3 3
%e ... 1 0 5* -> 2 1 1 -> 3 1 1 -> 3 2 1 -> 3 2 2
%e 5 4 3 5* 4 4 1 5* 5 2 1 6* 3 2 2
%e where * indicates the cell being processed.
%K nonn
%O 1,2
%A _Lars Blomberg_, Nov 07 2014
|
