A328506 Iteration of Abelian sandpile model where the n-th matrix expansions occurs. Begins with infinite sand in 1 X 1 matrix.

1, 5, 16, 36, 66, 101, 160, 218, 285, 374, 464, 565, 680, 815, 969, 1124, 1282, 1467, 1659, 1863, 2091, 2346, 2559, 2824, 3100, 3411, 3690, 4043, 4380, 4697, 5060, 5468, 5833, 6266, 6670, 7132, 7595, 8006, 8502, 9004, 9518, 10039, 10609, 11155, 11740, 12304, 12971, 13603, 14202, 14861, 15532, 16217

Offset

1,2

Comments

The Abelian sandpile model is a cellular automaton considering the behavior of sand grains on a square grid. Any square containing 4 or more grains will topple, sending a grain to each of its 4 neighbors and subtracting 4 grains from itself.

Here, expansion refers to the addition of a boundary layer to the outside of the existing matrix when the model reaches beyond the previous matrix boundary.

Links

Table of n, a(n) for n=1..52.

Example

                                                      _ _ _ _ _
          _ _ _      _ _ _      _ _ _      _ _ _     |0|0|1|0|0|
   _     |0|1|0|    |0|2|0|    |0|3|0|    |0|4|0|    |0|2|1|2|0|
  |∞| -> |1|∞|1| -> |2|∞|2| -> |3|∞|3| -> |4|∞|4| -> |1|1|∞|1|1| -> ...
   ‾     |0|1|0|    |0|2|0|    |0|3|0|    |0|4|0|    |0|2|1|2|0|
          ‾ ‾ ‾      ‾ ‾ ‾      ‾ ‾ ‾      ‾ ‾ ‾     |0|0|1|0|0|
                                                      ‾ ‾ ‾ ‾ ‾
            ^                                             ^
     1st expansion on                              2nd expansion on
   1st iteration (a(1) = 1)                      5th iteration (a(2) = 5)

Prog

(MATLAB)
L = 3;
plane = zeros(3, 3);
plane(2, 2) = 99999999999999999999999999999999999999999999999;
listn = [];
for n = 1:50000
    plane2 = plane;
    for r = 1:L
        for c = 1:L
            if plane(r, c) > 3
                plane2(r, c) = plane2(r, c) - 4;
                plane2(r-1, c) = plane2(r-1, c)+1;
                plane2(r+1, c) = plane2(r+1, c)+1;
                plane2(r, c-1) = plane2(r, c-1)+1;
                plane2(r, c+1) = plane2(r, c+1)+1;
            end
        end
    end
    if sum(plane2(:, 1))+sum(plane2(1, :)) > 0
        plane2 = padarray(plane2, [1, 1]);
        L = L+2;
        listn = [listn n];
    end
    plane = plane2;
end
fprintf('%s\n', sprintf('%d, ', listn))
(PARI)
Step(M)={my(n=#M, R=matrix(n, n)); for(i=2, n-1, for(j=2, n-1, if(M[i, j]>=4, R[i, j]-=4; R[i, j+1]++; R[i, j-1]++; R[i-1, j]++; R[i+1, j]++))); M+R}
Expand(M)={my(n=#M, R=matrix(n+2, n+2)); for(i=1, n, for(j=1, n, R[i+1, j+1]=M[i, j])); R}
seq(n)={my(L=List(), M=matrix(3, 3), k=0); while(#L<n, k++; my(o=#L+2); M[o, o]=4; M=Step(M); if(M[1, ], M=Expand(M); listput(L, k))); Vec(L)} \\ Andrew Howroyd, Oct 23 2019

Crossrefs

Cf. A259013, A249872.

Sequence in context: A108966 A321019 A184635 * A072333 A055232 A211806

Adjacent sequences: A328503 A328504 A328505 * A328507 A328508 A328509

Keyword

nonn

Author

Parker Grootenhuis, Oct 22 2019

Status

approved