A259013 a(n) is the smallest number of grains of sand placed at the center square of a (2n-1) X (2n-1) table so that some grains drop off the table by the end of the diffusion process.

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

Offset: 1

Author

Sezai ATA, Jun 16 2015

Keywords

• nonn

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

• Scott R. Shannon, Table of n, a(n) for n = 1..1000
• Wikipedia. Abelian Sandpile Model

Programs

MATLAB

% S(k) gives the minimum number of grains of sand needed at the center
% of a (2n-1) X (2n-1) 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*n-1;
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(xi-1,yi)=A(xi-1,yi) + 1;
                    A(xi,yi-1)=A(xi,yi-1) + 1;
                end
            end
        end
    end
    for k=1:n-1
        if A(n,n+k)==1 && firstsand(k)==0
        firstsand(k)=1;
        S(k)=j;
        end
    end
end

Extensions

a(21)-a(45) from Giovanni Resta, Jun 17 2015