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

%I A291896 #28 Mar 11 2022 08:00:58
%S A291896 1,1,1,2,3,5,9,14,24,40,67,112,186,312,520,868,1449,2417,4034,6730,
%T A291896 11229,18735,31254,52143,86989,145119,242096,403871,673751,1123964,
%U A291896 1875014,3127926,5218034,8704769,14521354,24224601,40411595,67414781,112461579,187608762
%N A291896 Number of 1-dimensional sandpiles with n grains piling up against the wall.
%C A291896 Number of compositions of n where the first part is 1 and the absolute difference between consecutive parts is <=1 (smooth compositions).
%H A291896 Seiichi Manyama, <a href="/A291896/b291896.txt">Table of n, a(n) for n = 0..4501</a>
%e A291896 The a(6)=9 smooth compositions of 6 are:
%e A291896 :
%e A291896 : oooooo|
%e A291896 :
%e A291896 :     o|
%e A291896 : ooooo|
%e A291896 :
%e A291896 :    o |
%e A291896 : ooooo|
%e A291896 :
%e A291896 :   o  |
%e A291896 : ooooo|
%e A291896 :
%e A291896 :  o   |
%e A291896 : ooooo|
%e A291896 :
%e A291896 :   oo|
%e A291896 : oooo|
%e A291896 :
%e A291896 :  o o|
%e A291896 : oooo|
%e A291896 :
%e A291896 :  oo |
%e A291896 : oooo|
%e A291896 :
%e A291896 :   o|
%e A291896 :  oo|
%e A291896 : ooo|
%p A291896 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A291896       add(b(n-j, j), j=max(1, i-1)..min(i+1, n)))
%p A291896     end:
%p A291896 a:= n-> b(n, 0):
%p A291896 seq(a(n), n=0..50);  # _Alois P. Heinz_, Sep 05 2017
%t A291896 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_ *)
%o A291896 (Python)
%o A291896 from sympy.core.cache import cacheit
%o A291896 @cacheit
%o A291896 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))
%o A291896 def a(n): return b(n, 0)
%o A291896 print([a(n) for n in range(51)]) # _Indranil Ghosh_, Sep 06 2017, after Maple code
%Y A291896 Cf. A186085, A291895.
%K A291896 nonn
%O A291896 0,4
%A A291896 _Seiichi Manyama_, Sep 05 2017