

A259013


a(n) is the smallest number of grains of sand placed at the center square of a (2n1) X (2n1) table so that some grains drop off the table by the end of the diffusion process.


7



4, 16, 44, 88, 144, 208, 320, 408, 512, 672, 788, 948, 1096, 1288, 1552, 1768, 1960, 2208, 2456, 2708, 3028, 3384, 3648, 3964, 4348, 4728, 5076, 5448, 5884, 6308, 6708, 7176, 7644, 8240, 8664, 9132, 9764, 10276, 10816, 11404, 11992, 12516, 13264, 13816, 14388
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OFFSET

1,1


COMMENTS

The diffusion rule is that if a square has more than 3 grains of sand then it loses 4 grains and each neighbor's number of grains increases by one. Initially the center square has a(n) sand grains and all other squares are empty. The final distribution of sand grains and the number a(n) do not depend on the order of the diffusion process. For this reason, it is called an "abelian sandpile model".


LINKS



PROG

(MATLAB)
% S(k) gives the minimum number of grains of sand needed at the center
% of a (2n1) X (2n1) square table for some grains to drop off
% the table in an "abelian sandpile model".
firstsand=zeros(1, 49);
S=zeros(1, 49);
n=50;
lim=2*n1;
A=zeros(lim, lim);
for j=1:17128;
A(n, n)= A(n, n)+1;
while max(max(A))>=4
for xi=1:lim
for yi=1:lim
if A(xi, yi) >= 4
A(xi, yi)= A(xi, yi)  4;
A(xi+1, yi)=A(xi+1, yi) + 1;
A(xi, yi+1)=A(xi, yi+1) + 1;
A(xi1, yi)=A(xi1, yi) + 1;
A(xi, yi1)=A(xi, yi1) + 1;
end
end
end
end
for k=1:n1
if A(n, n+k)==1 && firstsand(k)==0
firstsand(k)=1;
S(k)=j;
end
end
end


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



