A224960 Number of compositions [p(1), p(2), ..., p(k)] of n such that p(j) >= p(1) - 1.

1, 1, 2, 4, 7, 14, 26, 52, 101, 200, 396, 787, 1564, 3117, 6214, 12398, 24749, 49427, 98740, 197303, 394323, 788201, 1575695, 3150265, 6298732, 12594595, 25184598, 50361842, 100711888, 201404839, 402779246, 805509560, 1610940381, 3221753990

Offset: 0

Author

Joerg Arndt, Apr 21 2013

Keywords

• nonn

Examples

			The a(5) = 14 such compositions of 5 are
01:  [ 1 1 1 1 1 ]
02:  [ 1 1 1 2 ]
03:  [ 1 1 2 1 ]
04:  [ 1 1 3 ]
05:  [ 1 2 1 1 ]
06:  [ 1 2 2 ]
07:  [ 1 3 1 ]
08:  [ 1 4 ]
09:  [ 2 1 1 1 ]
10:  [ 2 1 2 ]
11:  [ 2 2 1 ]
12:  [ 2 3 ]
13:  [ 3 2 ]
14:  [ 5 ]
(the two forbidden compositions are [ 3 1 1 ] and [ 4 1 ]).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A171682 (compositions such that p(j) >= p(1)).

Cf. A079501 (compositions such that p(j) > p(1)).

Cf. A048888 (compositions such that p(j) <= p(1) + 1).

Cf. A007059 (compositions such that p(j) < p(1)).

Cf. A079500 (compositions such that p(j) <= p(1)).

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, add(b(n-j,
      `if`(i=0, max(1, j-1), i)), j=`if`(i=0, 1, i)..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, May 02 2013

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n - j, If[i == 0, Max[1, j - 1], i]], {j, If[i == 0, 1, i], n}]];
a[n_] := b[n, 0];
a /@ Range[0, 50] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

Formula

a(n) ~ 3 * 2^(n-3). - Vaclav Kotesovec, May 01 2014