A224957 Number of compositions [p(1), p(2), ..., p(k)] of n such that p(j) <= 2*p(j-1) and p(j-1) <= 2*p(j).

1, 1, 2, 4, 6, 11, 19, 31, 54, 92, 154, 266, 454, 771, 1319, 2249, 3834, 6550, 11176, 19069, 32558, 55567, 94838, 161891, 276325, 471659, 805102, 1374234, 2345724, 4004031, 6834605, 11666260, 19913668, 33991462, 58021534, 99039592, 169055094, 288567886, 492569833, 840790082

Offset: 0

Author

Joerg Arndt, Apr 21 2013

Keywords

• nonn

Examples

			There are a(6) = 19 such compositions of 6:
01:  [ 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 2 ]
03:  [ 1 1 1 2 1 ]
04:  [ 1 1 2 1 1 ]
05:  [ 1 1 2 2 ]
06:  [ 1 2 1 1 1 ]
07:  [ 1 2 1 2 ]
08:  [ 1 2 2 1 ]
09:  [ 1 2 3 ]
10:  [ 2 1 1 1 1 ]
11:  [ 2 1 1 2 ]
12:  [ 2 1 2 1 ]
13:  [ 2 2 1 1 ]
14:  [ 2 2 2 ]
15:  [ 2 4 ]
16:  [ 3 2 1 ]
17:  [ 3 3 ]
18:  [ 4 2 ]
19:  [ 6 ]
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..4306

Crossrefs

The case of strict relations is A342330, with strict case A342341.

The strict case is A342342.

A000929 counts partitions with adjacent parts x >= 2y.

A002843 counts compositions with adjacent parts x <= 2y.

A045690 counts sets with maximum n with adjacent elements y < 2x.

A154402 counts partitions with adjacent parts x = 2y.

A274199 counts compositions with adjacent parts x < 2y.

A342094 counts partitions with adjacent parts x <= 2y (strict: A342095).

A342096 counts partitions without adjacent x >= 2y (strict: A342097).

A342098 counts partitions with adjacent parts x > 2y.

A342331 counts compositions with adjacent parts x = 2y or y = 2x.

A342332 counts compositions with adjacent parts x > 2y or y > 2x.

A342333 counts compositions with adjacent parts x >= 2y or y >= 2x.

A342334 counts compositions with adjacent parts x >= 2y or y > 2x.

A342335 counts compositions with adjacent parts x >= 2y or y = 2x.

A342336 counts compositions with adjacent parts x > 2y or y = 2x.

A342337 counts partitions with adjacent parts x = y or x = 2y.

A342338 counts compositions with adjacent parts x < 2y and y <= 2x.

A342340 counts compositions with adjacent x = y or x = 2y or y = 2x.

Cf. A003114, A003242, A034296, A167606, A342191, A342339.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, add(
      b(n-j, j), j=`if`(i=0, 1..n, ceil(i/2)..min(n, 2*i))))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..42);  # Alois P. Heinz, Mar 15 2021

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Table[#[[i]]<=2*#[[i-1]]&&#[[i-1]]<=2*#[[i]],{i,2,Length[#]}]&]],{n,15}] (* Gus Wiseman, Mar 12 2021 *)
b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n - j, j], {j, If[i == 0, Range[n], Range[Ceiling[i/2], Min[n, 2*i]]]}]];
a[n_] := b[n, 0];
a /@ Range[0, 42] (* Jean-François Alcover, May 24 2021, after Alois P. Heinz *)

Extensions

Name corrected by Gus Wiseman, Mar 11 2021