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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.

%I A180230 #17 Dec 18 2021 08:04:59
%S A180230 2,6,10,22,26,50,66,78,122,142,154,194,254,270,342,386,418,490,518,
%T A180230 578,654,698,766,914,942,1074,1150,1178,1310,1366,1410,1570,1646,1794,
%U A180230 1894,2054,2130,2246,2406,2466,2654,2742,2894,3006,3138,3318,3582,3670,3826
%N 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.
%C A180230 The abelian sandpile growth model starts with height h on every site of the square grid.
%C A180230 An addition increases the height of the origin by 1. After each addition, the model is stabilized by toppling unstable sites.
%C A180230 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.
%C A180230 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.
%C A180230 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.
%H A180230 Anne Fey, <a href="/A180230/a180230.txt">MATLAB program</a>
%H A180230 Anne Fey, Lionel Levine and Yuval Peres, <a href="https://arxiv.org/abs/0901.3805">Growth rates and explosions in sandpiles</a>, arXiv:0901.3805 [math.CO], 2009.
%H A180230 Anne Fey, Lionel Levine and Yuval Peres, <a href="https://doi.org/10.1007/s10955-009-9899-6">Growth Rates and Explosions in Sandpiles</a>, Journal of Statistical Physics, Vol. 138, No. 1-3 (2010), 143-159.
%H A180230 Anne Fey and Frank Redig, <a href="https://arxiv.org/abs/math/0702450">Limiting shapes for deterministic centrally seeded growth models</a>, arXiv:math/0702450 [math.PR], 2007.
%H A180230 Anne Fey and Frank Redig, <a href="https://doi.org/10.1007/s10955-007-9450-6">Limiting Shapes for Deterministic Centrally Seeded Growth Models</a>, Journal of Statistical Physics 130 (2008), 579-597.
%H A180230 Rémy Sigrist, <a href="/A180230/a180230_1.txt">C++ program for A180230</a>
%e A180230 After 2 additions, the origin is unstable and topples once. Then every site is stable. Therefore a(0)=2.
%e A180230 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.
%o A180230 (C++) See Links section.
%Y A180230 Cf. A056219
%K A180230 nonn
%O A180230 0,1
%A A180230 Anne Fey (a.c.fey-denboer(AT)tudelft.nl), Aug 17 2010
%E A180230 More terms from _Rémy Sigrist_, Dec 15 2021