OFFSET
0,1
COMMENTS
The abelian sandpile growth model starts with height h on every site of the square grid.
An addition increases the height of the origin by 1. After each addition, the model is stabilized by toppling unstable sites.
A site is unstable if its height is at least 4; in a toppling, its height decreases by 4 and the height of its neighbors increases by 1.
If h=2, then for any number of additions, the set of sites that toppled at least once is a square. This was proved in Fey-Redig-2008.
For all n, a(n) <= (2n+3)^2. In Fey-Levine-Peres-2010, it was proved that for n large enough, a(n) >= Pi/4 n^2.
LINKS
Anne Fey, MATLAB program
Anne Fey, Lionel Levine and Yuval Peres, Growth rates and explosions in sandpiles, arXiv:0901.3805 [math.CO], 2009.
Anne Fey, Lionel Levine and Yuval Peres, Growth Rates and Explosions in Sandpiles, Journal of Statistical Physics, Vol. 138, No. 1-3 (2010), 143-159.
Anne Fey and Frank Redig, Limiting shapes for deterministic centrally seeded growth models, arXiv:math/0702450 [math.PR], 2007.
Anne Fey and Frank Redig, Limiting Shapes for Deterministic Centrally Seeded Growth Models, Journal of Statistical Physics 130 (2008), 579-597.
Rémy Sigrist, C++ program for A180230
EXAMPLE
After 2 additions, the origin is unstable and topples once. Then every site is stable. Therefore a(0)=2.
After 4 more additions, the origin topples again. Then more sites become unstable, so that the set of sites that toppled at least once becomes the square with radius 1. Therefore a(1) = 6.
PROG
(C++) See Links section.
CROSSREFS
KEYWORD
nonn
AUTHOR
Anne Fey (a.c.fey-denboer(AT)tudelft.nl), Aug 17 2010
EXTENSIONS
More terms from Rémy Sigrist, Dec 15 2021
STATUS
approved