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
Examples
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
Links
- Seiichi Manyama, Rows n = 0..25, flattened
Programs
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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 *)
Formula
T(n,1) + T(n,2) = Fibonacci(n).
Comments