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

0,18

Comments

See A186085 for the definition of sandpiles.

Links

Table of n, a(n) for n=0..125.

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

Adjacent sequences: A187078 A187079 A187080 * A187082 A187083 A187084

Keyword

nonn,tabl

Author

Joerg Arndt, Mar 08 2011

Status

approved