A261983 Number of compositions of n such that at least two adjacent parts are equal.

0, 0, 1, 1, 4, 9, 18, 41, 89, 185, 388, 810, 1670, 3435, 7040, 14360, 29226, 59347, 120229, 243166, 491086, 990446, 1995410, 4016259, 8076960, 16231746, 32599774, 65437945, 131293192, 263316897, 527912140, 1058061751, 2120039885, 4246934012, 8505864640

Offset: 0

Author

Alois P. Heinz, Sep 07 2015

Keywords

• nonn

Examples

			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)
		

Links

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

Crossrefs

Column k=1 of A261981.

Cf. A011782, A262046.

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.

Cf. A114901, A178470, A242882, A244164, A325534, A335448, A335452.

Programs

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 *)

Formula

a(n) ~ 2^(n-1). - Vaclav Kotesovec, Sep 08 2015

a(n) = A011782(n) - A003242(n). - Emeric Deutsch, Jul 03 2020