OFFSET
0,5
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
FORMULA
a(n) ~ 2^(n-1). - Vaclav Kotesovec, Sep 08 2015
EXAMPLE
a(5) = 9: 311, 113, 221, 122, 2111, 1211, 1121, 1112, 11111.
From Gus Wiseman, Jul 07 2020: (Start)
The a(2) = 1 through a(6) = 18 compositions:
(1,1) (1,1,1) (2,2) (1,1,3) (3,3)
(1,1,2) (1,2,2) (1,1,4)
(2,1,1) (2,2,1) (2,2,2)
(1,1,1,1) (3,1,1) (4,1,1)
(1,1,1,2) (1,1,1,3)
(1,1,2,1) (1,1,2,2)
(1,2,1,1) (1,1,3,1)
(2,1,1,1) (1,2,2,1)
(1,1,1,1,1) (1,3,1,1)
(2,1,1,2)
(2,2,1,1)
(3,1,1,1)
(1,1,1,1,2)
(1,1,1,2,1)
(1,1,2,1,1)
(1,2,1,1,1)
(2,1,1,1,1)
(1,1,1,1,1,1)
(End)
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 0, add(
`if`(i=j, ceil(2^(n-j-1)), b(n-j, j)), j=1..n))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..40);
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], MatchQ[#, {___, x_, x_, ___}]&]], {n, 0, 10}] (* Gus Wiseman, Jul 06 2020 *)
b[n_, i_] := b[n, i] = If[n == 0, 0, Sum[If[i == j, Ceiling[2^(n-j-1)], b[n-j, j]], {j, 1, n}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 20 2023, after Alois P. Heinz's Maple code *)
CROSSREFS
Column k=1 of A261981.
The complement A003242 counts anti-runs.
Sum of positive-indexed terms of row n of A106356.
Row sums of A131044.
The (1,1,1) matching case is A335464.
Strict compositions are A032020.
Compositions with adjacent parts coprime are A167606.
Compositions with equal parts contiguous are A274174.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 07 2015
STATUS
approved