A224958 Number of compositions [p(1), p(2), ..., p(k)] of n such that p(j) != p(j-2)

1, 1, 2, 3, 6, 9, 18, 29, 53, 91, 162, 277, 495, 855, 1508, 2625, 4618, 8049, 14130, 24675, 43255, 75621, 132475, 231697, 405751, 709887, 1242824, 2174763, 3806989, 6662291, 11661737, 20409409, 35723307, 62521919, 109431810, 191527623, 335225350, 586717615

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..500

Formula

a(n) ~ c * d^n, where d = 1.7502412917183090312497386246... (see A241902) and c = 0.5940298439978189763822100914... - Vaclav Kotesovec, May 01 2014

Example

The a(6) = 18 such compositions of 6 are
01:  [ 1 1 2 2 ]
02:  [ 1 1 4 ]
03:  [ 1 2 2 1 ]
04:  [ 1 2 3 ]
05:  [ 1 3 2 ]
06:  [ 1 5 ]
07:  [ 2 1 1 2 ]
08:  [ 2 1 3 ]
09:  [ 2 2 1 1 ]
10:  [ 2 3 1 ]
11:  [ 2 4 ]
12:  [ 3 1 2 ]
13:  [ 3 2 1 ]
14:  [ 3 3 ]
15:  [ 4 1 1 ]
16:  [ 4 2 ]
17:  [ 5 1 ]
18:  [ 6 ]

Maple

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

Mathematica

b[n_, i_, j_] := b[n, i, j] = If[n==0, 1, Sum[If[k==j, 0, b[n-k, If[n-k < k, 0, k], If[n-k < i, 0, i]]], {k, 1, n}]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 08 2015, after Alois P. Heinz *)

Crossrefs

Cf. A000726 (partitions such that p(j) != p(j-2)), A003242, A241902.

Sequence in context: A165647 A191398 A066313 * A304912 A018499 A107847

Adjacent sequences: A224955 A224956 A224957 * A224959 A224960 A224961

Keyword

nonn

Author

Joerg Arndt, Apr 21 2013

Status

approved