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A180230 a(n) is the minimal number of additions needed to grow to radius n, in the two-dimensional abelian sandpile growth model with h=2. 1
2, 6, 10, 22, 26, 50, 66, 78, 122, 142, 154, 194, 254, 270, 342, 386, 418, 490, 518, 578, 654, 698, 766, 914, 942, 1074, 1150, 1178, 1310, 1366, 1410, 1570, 1646, 1794, 1894 (list; graph; refs; listen; history; text; internal format)



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.


Anne Fey, Lionel Levine and Yuval Peres, Growth rates and explosions in sandpiles, Journal of Statistical Physics 138 (2010), 143-159.

Anne Fey and Frank Redig, Limiting shapes for deterministic centrally seeded growth models , Journal of Statistical Physics 130 (2008), 579-597.


Table of n, a(n) for n=0..34.

Anne Fey, MATLAB program


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.


Cf. A056219

Sequence in context: A272952 A034450 A112861 * A186296 A140775 A077064

Adjacent sequences:  A180227 A180228 A180229 * A180231 A180232 A180233




Anne Fey (a.c.fey-denboer(AT)tudelft.nl), Aug 17 2010



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Last modified November 20 08:12 EST 2017. Contains 294962 sequences.