A293452
Triangle T(n,k) read by rows: T(n,k) is the number of iterations to reach a final state for an n X k lattice of sandpiles on a torus according to rules specified in A249872.
Original entry on oeis.org
0, 1, 7, 2, 14, 28, 7, 35, 65, 133, 10, 47, 86, 198, 316, 22, 86, 134, 331, 487, 913, 28, 106, 164, 399, 696, 1099, 1360, 50, 159, 288, 589, 930, 1518, 1798, 2987, 60, 187, 336, 681, 1070, 1966, 2320, 3432, 4340, 95, 265, 515, 1052, 1386, 2430, 3475, 4484, 5977, 7495, 110, 303, 584, 1184, 1556, 2718
Offset: 1
Triangle begins:
0
1, 7
2, 14, 28
7, 35, 65, 133
10, 47, 86, 198, 316
22, 86, 134, 331, 487, 913
28, 106, 164, 399, 696, 1099, 1360
50, 159, 288, 589, 930, 1518, 1798, 2987
60, 187, 336, 681, 1070, 1966, 2320, 3432, 4340
95, 265, 515, 1052, 1386, 2430, 3475, 4484, 5977, 7495
...
A328506
Iteration of Abelian sandpile model where the n-th matrix expansions occurs. Begins with infinite sand in 1 X 1 matrix.
Original entry on oeis.org
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
_ _ _ _ _
_ _ _ _ _ _ _ _ _ _ _ _ |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)
-
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))
-
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(#LAndrew Howroyd, Oct 23 2019
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