A291896 Number of 1-dimensional sandpiles with n grains piling up against the wall.

1, 1, 1, 2, 3, 5, 9, 14, 24, 40, 67, 112, 186, 312, 520, 868, 1449, 2417, 4034, 6730, 11229, 18735, 31254, 52143, 86989, 145119, 242096, 403871, 673751, 1123964, 1875014, 3127926, 5218034, 8704769, 14521354, 24224601, 40411595, 67414781, 112461579, 187608762

Offset: 0

Author

Seiichi Manyama, Sep 05 2017

Keywords

• nonn

Comments

Number of compositions of n where the first part is 1 and the absolute difference between consecutive parts is <=1 (smooth compositions).

Examples

			The a(6)=9 smooth compositions of 6 are:
:
: oooooo|
:
:     o|
: ooooo|
:
:    o |
: ooooo|
:
:   o  |
: ooooo|
:
:  o   |
: ooooo|
:
:   oo|
: oooo|
:
:  o o|
: oooo|
:
:  oo |
: oooo|
:
:   o|
:  oo|
: ooo|
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..4501

Crossrefs

Cf. A186085, A291895.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
      add(b(n-j, j), j=max(1, i-1)..min(i+1, n)))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 05 2017

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, Sum[b[n-j, j], {j, Max[1, i-1], Min[i+1, n]}]]; a[n_] := b[n, 0]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, May 29 2019, after Alois P. Heinz *)

Python

from sympy.core.cache import cacheit
@cacheit
def b(n, i): return 1 if n==0 else sum(b(n - j, j) for j in range(max(1, i - 1), min(i + 1, n) + 1))
def a(n): return b(n, 0)
print([a(n) for n in range(51)]) # Indranil Ghosh, Sep 06 2017, after Maple code