A187080 Triangle T(n,k) read by rows: fountains of n coins and height k.

1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 4, 0, 0, 0, 0, 1, 7, 1, 0, 0, 0, 0, 1, 12, 2, 0, 0, 0, 0, 0, 1, 20, 5, 0, 0, 0, 0, 0, 0, 1, 33, 11, 0, 0, 0, 0, 0, 0, 0, 1, 54, 22, 1, 0, 0, 0, 0, 0, 0, 0, 1, 88, 44, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 143, 85, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 232, 161, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 376, 302, 25, 0, 0, 0

Offset

0,13

Comments

See A005169 for the definition of a "fountain of n coins". [John W. Layman, Mar 10 2011]

Links

Seiichi Manyama, Rows n = 0..25, flattened

Formula

T(n,1) + T(n,2) = Fibonacci(n).

Example

Triangle begins:
1;
0,1;
0,1,0;
0,1,1,0;
0,1,2,0,0;
0,1,4,0,0,0;
0,1,7,1,0,0,0;
0,1,12,2,0,0,0,0;
0,1,20,5,0,0,0,0,0;
0,1,33,11,0,0,0,0,0,0;
0,1,54,22,1,0,0,0,0,0,0;
0,1,88,44,2,0,0,0,0,0,0,0;
0,1,143,85,5,0,0,0,0,0,0,0,0;
0,1,232,161,12,0,0,0,0,0,0,0,0,0;
0,1,376,302,25,0,0,0,0,0,0,0,0,0,0;
0,1,609,559,52,1,0,0,0,0,0,0,0,0,0,0;
0,1,986,1026,105,2,0,0,0,0,0,0,0,0,0,0,0;
0,1,1596,1870,207,5,0,0,0,0,0,0,0,0,0,0,0,0;
The 15 compositions corresponding to fountains of 7 coins are the following:
   #:    composition      height
   1:    [ 1 2 3 1 ]        3
   2:    [ 1 2 2 2 ]        2
   3:    [ 1 1 2 3 ]        3
   4:    [ 1 2 2 1 1 ]      2
   5:    [ 1 2 1 2 1 ]      2
   6:    [ 1 1 2 2 1 ]      2
   7:    [ 1 2 1 1 2 ]      2
   8:    [ 1 1 2 1 2 ]      2
   9:    [ 1 1 1 2 2 ]      2
  10:    [ 1 2 1 1 1 1 ]    2
  11:    [ 1 1 2 1 1 1 ]    2
  12:    [ 1 1 1 2 1 1 ]    2
  13:    [ 1 1 1 1 2 1 ]    2
  14:    [ 1 1 1 1 1 2 ]    2
  15:    [ 1 1 1 1 1 1 1 ]  1
  stats:  0 1 12 2 0 0 0 0

Mathematica

b[n_, i_, h_] := b[n, i, h] = If[n == 0, x^h, Sum[b[n - j, j, Max[h, j]], {j, 1, Min[i + 1, n]}]];
T[n_] := Table[Coefficient[#, x, i], {i, 0, n}]& @ b[n, 0, 0];
Table[T[n], {n, 0, 25}] // Flatten (* Jean-François Alcover, May 31 2019, after Alois P. Heinz in A291878 *)

Crossrefs

Row sums give A005169 (fountains of n coins).

Cf. A047998, A187081 (sandpiles by height).

Sequence in context: A123262 A191906 A371647 * A301342 A226369 A320751

Adjacent sequences: A187077 A187078 A187079 * A187081 A187082 A187083

Keyword

nonn,tabl

Author

Joerg Arndt, Mar 08 2011

Status

approved