A224959 Number of compositions [p(1), p(2), ..., p(k)] of n such that p(j) - p(j-1) <= 2

1, 1, 2, 4, 8, 15, 29, 55, 105, 199, 378, 716, 1358, 2572, 4873, 9229, 17480, 33102, 62688, 118709, 224795, 425676, 806068, 1526371, 2890338, 5473125, 10363871, 19624925, 37161558, 70368705, 133249369, 252319408, 477788980, 904735349, 1713195705, 3244086145

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..3607

Formula

a(n) ~ c * d^n, where d=1.893587506319686491635881459546948770530553555112342985931092896452453511... and c=0.6398882559654423774981963082429746674258714212085034829366885993226... - Vaclav Kotesovec, May 01 2014

Example

There are a(5) = 15 such compositions of 5:
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:  [ 2 1 1 1 ]
09:  [ 2 1 2 ]
10:  [ 2 2 1 ]
11:  [ 2 3 ]
12:  [ 3 1 1 ]
13:  [ 3 2 ]
14:  [ 4 1 ]
15:  [ 5 ]
(the single forbidden composition is [ 1 4 ]).

Maple

b:= proc(n, i) option remember;
      `if`(n=0, 1, add(b(n-j, max(1, j-2)), j=i..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..40);  # Alois P. Heinz, May 02 2013

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, Sum[b[n-j, Max[1, j-2]], {j, i, n}]];
a[n_] := b[n, 1];
a /@ Range[0, 40] (* Jean-François Alcover, Dec 19 2020, after Alois P. Heinz *)

Crossrefs

Cf. A003116 (compositions such that p(j) - p(j-1) <= 1).

Cf. A225084 (triangle: compositions of n such that max(p(j) - p(j-1)) = k).

Cf. A225085 (triangle: compositions of n such that max(p(j) - p(j-1)) <= k).

Sequence in context: A208976 A278554 A335473 * A108564 A066369 A239555

Adjacent sequences: A224956 A224957 A224958 * A224960 A224961 A224962

Keyword

nonn

Author

Joerg Arndt, Apr 21 2013

Status

approved