A187081 Triangle T(n,k) read by rows: sandpiles of n grains and height k.

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

Author

Joerg Arndt, Mar 08 2011

Keywords

• nonn
• tabl

Comments

See A186085 for the definition of sandpiles.

Examples

			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).

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]