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

0,7

Comments

Number of compositions of n where the first part is 1 and the absolute difference between consecutive parts is 1.

Links

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

Example

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|

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 *)

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) 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

Crossrefs

Cf. A291896, A291904.

Sequence in context: A291874 A049346 A227310 * A347584 A366398 A365932

Adjacent sequences: A291902 A291903 A291904 * A291906 A291907 A291908

Keyword

nonn

Author

Seiichi Manyama, Sep 05 2017

Status

approved