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.

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, 2054, 2130, 2246, 2406, 2466, 2654, 2742, 2894, 3006, 3138, 3318, 3582, 3670, 3826

Offset: 0

Author

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

Keywords

• nonn

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.

Examples

			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.
		

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

Crossrefs

Cf. A056219

Extensions

More terms from Rémy Sigrist, Dec 15 2021