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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 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

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