

A180230


a(n) is the minimal number of additions needed to grow to radius n, in the twodimensional abelian sandpile growth model with h=2.


7



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
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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 FeyRedig2008.
For all n, a(n) <= (2n+3)^2. In FeyLevinePeres2010, it was proved that for n large enough, a(n) >= Pi/4 n^2.


LINKS

Table of n, a(n) for n=0..48.
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. 13 (2010), 143159.
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), 579597.
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

Cf. A056219
Sequence in context: A297185 A304991 A112861 * A186296 A140775 A077064
Adjacent sequences: A180227 A180228 A180229 * A180231 A180232 A180233


KEYWORD

nonn


AUTHOR

Anne Fey (a.c.feydenboer(AT)tudelft.nl), Aug 17 2010


EXTENSIONS

More terms from Rémy Sigrist, Dec 15 2021


STATUS

approved



