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A328506 Iteration of Abelian sandpile model where the n-th matrix expansions occurs. Begins with infinite sand in 1 X 1 matrix. 0
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified April 6 11:02 EDT 2020. Contains 333273 sequences. (Running on oeis4.)