cp's OEIS Frontend

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.

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