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

0,4

Comments

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

Links

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

Example

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|

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 *)

Prog

(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

Crossrefs

Cf. A186085, A291895.

Sequence in context: A026746 A004699 A245800 * A018155 A227375 A032089

Adjacent sequences: A291893 A291894 A291895 * A291897 A291898 A291899

Keyword

nonn

Author

Seiichi Manyama, Sep 05 2017

Status

approved