A291905 Row sums of A291904.

1, 1, 0, 1, 1, 0, 2, 1, 1, 3, 2, 3, 4, 4, 6, 8, 8, 11, 14, 16, 21, 26, 32, 39, 49, 60, 75, 93, 114, 142, 176, 217, 268, 334, 411, 510, 632, 779, 967, 1196, 1477, 1832, 2266, 2801, 3470, 4291, 5310, 6572, 8129, 10061, 12449, 15401, 19058, 23581, 29178, 36102, 44668

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.

Examples

			The a(6)=2 compositions of 6 are:
:
:  o o|
: oooo|
:
:   o|
:  oo|
: ooo|
:
The a(9)=3 compositions of 9 are:
:
:   o  |
:  ooo |
: ooooo|
:
:  o o o|
: oooooo|
:
:     o|
:  o oo|
: ooooo|
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A291896, A291904.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, add(
     `if`(j=i, 0, b(n-j, j)), j=max(1, i-1)..min(i+1, n)))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 05 2017

Mathematica

T[0, 0] = 1; T[, 0] = 0; T[n?Positive, k_] /; 0 < k <= Floor[(Sqrt[8n+1] - 1)/2] := T[n, k] = T[n-k, k-1] + T[n-k, k+1]; T[, ] = 0;
a[n_] := Sum[T[n, k], {k, 0, Floor[(Sqrt[8n+1] - 1)/2]}];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 29 2019 *)

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) if j != i)
def a(n): return b(n, 0)
print([a(n) for n in range(61)]) # Indranil Ghosh, Sep 06 2017, after Maple program