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A187081
Triangle T(n,k) read by rows: sandpiles of n grains and height k.
1
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, 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, 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
OFFSET
0,18
COMMENTS
See A186085 for the definition of sandpiles.
FORMULA
For n>=2 we have T(n,1)+T(n,2) = Fibonacci(n-1).
T(n,2) = A000071(n). [Joerg Arndt, Sep 17 2013]
EXAMPLE
Triangle begins:
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;
0,1,7,0,0,0,0,0;
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;
0,1,54,5,0,0,0,0,0,0,0,0;
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;
0,1,376,84,0,0,0,0,0,0,0,0,0,0,0,0;
0,1,609,158,1,0,0,0,0,0,0,0,0,0,0,0,0;
0,1,986,293,2,0,0,0,0,0,0,0,0,0,0,0,0,0;
0,1,1596,535,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0;
0,1,2583,969,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;
0,1,4180,1739,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;
0,1,6764,3099,52,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;
0,1,10945,5491,103,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;
The 22 compositions corresponding to sandpiles of 9 grains are the following:
#: composition height
1: [ 1 2 3 2 1 ] 3
2: [ 1 2 2 2 1 1 ] 2
3: [ 1 2 2 1 2 1 ] 2
4: [ 1 2 1 2 2 1 ] 2
5: [ 1 1 2 2 2 1 ] 2
6: [ 1 2 2 1 1 1 1 ] 2
7: [ 1 2 1 2 1 1 1 ] 2
8: [ 1 1 2 2 1 1 1 ] 2
9: [ 1 2 1 1 2 1 1 ] 2
10: [ 1 1 2 1 2 1 1 ] 2
11: [ 1 1 1 2 2 1 1 ] 2
12: [ 1 2 1 1 1 2 1 ] 2
13: [ 1 1 2 1 1 2 1 ] 2
14: [ 1 1 1 2 1 2 1 ] 2
15: [ 1 1 1 1 2 2 1 ] 2
16: [ 1 2 1 1 1 1 1 1 ] 2
17: [ 1 1 2 1 1 1 1 1 ] 2
18: [ 1 1 1 2 1 1 1 1 ] 2
19: [ 1 1 1 1 2 1 1 1 ] 2
20: [ 1 1 1 1 1 2 1 1 ] 2
21: [ 1 1 1 1 1 1 2 1 ] 2
22: [ 1 1 1 1 1 1 1 1 1 ] 1
stats: 0 1 20 1 0 0 0 0 0 0
CROSSREFS
Row sums are A186085 (sandpiles of n grains), cf. A186084 (sandpiles by base length), A047998 (fountains of coins by base length).
Sequence in context: A364045 A166301 A354059 * A369308 A212434 A227186
KEYWORD
nonn,tabl
AUTHOR
Joerg Arndt, Mar 08 2011
STATUS
approved