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%I #23 Mar 11 2022 07:45:57
%S 1,0,1,0,1,0,0,1,0,0,0,1,1,0,0,0,1,2,0,0,0,0,1,4,0,0,0,0,0,1,7,0,0,0,
%T 0,0,0,1,12,0,0,0,0,0,0,0,1,20,1,0,0,0,0,0,0,0,1,33,2,0,0,0,0,0,0,0,0,
%U 1,54,5,0,0,0,0,0,0,0,0,0,1,88,11,0,0,0,0,0,0,0,0,0,0,1,143,22,0,0,0,0,0,0,0,0,0,0,0,1,232,44,0,0,0,0,0,0,0,0,0,0,0,0,1,376,84,0,0
%N Triangle T(n,k) read by rows: sandpiles of n grains and height k.
%C See A186085 for the definition of sandpiles.
%F For n>=2 we have T(n,1)+T(n,2) = Fibonacci(n-1).
%F T(n,2) = A000071(n). [_Joerg Arndt_, Sep 17 2013]
%e Triangle begins:
%e 1;
%e 0,1;
%e 0,1,0;
%e 0,1,0,0;
%e 0,1,1,0,0;
%e 0,1,2,0,0,0;
%e 0,1,4,0,0,0,0;
%e 0,1,7,0,0,0,0,0;
%e 0,1,12,0,0,0,0,0,0;
%e 0,1,20,1,0,0,0,0,0,0;
%e 0,1,33,2,0,0,0,0,0,0,0;
%e 0,1,54,5,0,0,0,0,0,0,0,0;
%e 0,1,88,11,0,0,0,0,0,0,0,0,0;
%e 0,1,143,22,0,0,0,0,0,0,0,0,0,0;
%e 0,1,232,44,0,0,0,0,0,0,0,0,0,0,0;
%e 0,1,376,84,0,0,0,0,0,0,0,0,0,0,0,0;
%e 0,1,609,158,1,0,0,0,0,0,0,0,0,0,0,0,0;
%e 0,1,986,293,2,0,0,0,0,0,0,0,0,0,0,0,0,0;
%e 0,1,1596,535,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0;
%e 0,1,2583,969,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;
%e 0,1,4180,1739,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;
%e 0,1,6764,3099,52,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;
%e 0,1,10945,5491,103,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;
%e The 22 compositions corresponding to sandpiles of 9 grains are the following:
%e #: composition height
%e 1: [ 1 2 3 2 1 ] 3
%e 2: [ 1 2 2 2 1 1 ] 2
%e 3: [ 1 2 2 1 2 1 ] 2
%e 4: [ 1 2 1 2 2 1 ] 2
%e 5: [ 1 1 2 2 2 1 ] 2
%e 6: [ 1 2 2 1 1 1 1 ] 2
%e 7: [ 1 2 1 2 1 1 1 ] 2
%e 8: [ 1 1 2 2 1 1 1 ] 2
%e 9: [ 1 2 1 1 2 1 1 ] 2
%e 10: [ 1 1 2 1 2 1 1 ] 2
%e 11: [ 1 1 1 2 2 1 1 ] 2
%e 12: [ 1 2 1 1 1 2 1 ] 2
%e 13: [ 1 1 2 1 1 2 1 ] 2
%e 14: [ 1 1 1 2 1 2 1 ] 2
%e 15: [ 1 1 1 1 2 2 1 ] 2
%e 16: [ 1 2 1 1 1 1 1 1 ] 2
%e 17: [ 1 1 2 1 1 1 1 1 ] 2
%e 18: [ 1 1 1 2 1 1 1 1 ] 2
%e 19: [ 1 1 1 1 2 1 1 1 ] 2
%e 20: [ 1 1 1 1 1 2 1 1 ] 2
%e 21: [ 1 1 1 1 1 1 2 1 ] 2
%e 22: [ 1 1 1 1 1 1 1 1 1 ] 1
%e stats: 0 1 20 1 0 0 0 0 0 0
%Y Row sums are A186085 (sandpiles of n grains), cf. A186084 (sandpiles by base length), A047998 (fountains of coins by base length).
%K nonn,tabl
%O 0,18
%A _Joerg Arndt_, Mar 08 2011
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