A335464 Number of compositions of n with a run of length > 2.

0, 0, 0, 1, 1, 3, 8, 18, 39, 86, 188, 406, 865, 1836, 3874, 8135, 17003, 35413, 73516, 152171, 314151, 647051, 1329936, 2728341, 5587493, 11424941, 23327502, 47567628, 96879029, 197090007, 400546603, 813258276, 1649761070, 3343936929, 6772740076, 13707639491

Offset: 0

Author

Gus Wiseman, Jul 06 2020

Keywords

• nonn

Comments

A composition of n is a finite sequence of positive integers summing to n.

Also compositions contiguously matching the pattern (1,1,1).

Examples

			The a(3) = 1 through a(7) = 18 compositions:
  (111)  (1111)  (1112)   (222)     (1114)
                 (2111)   (1113)    (1222)
                 (11111)  (3111)    (2221)
                          (11112)   (4111)
                          (11121)   (11113)
                          (12111)   (11122)
                          (21111)   (11131)
                          (111111)  (13111)
                                    (21112)
                                    (22111)
                                    (31111)
                                    (111112)
                                    (111121)
                                    (111211)
                                    (112111)
                                    (121111)
                                    (211111)
                                    (1111111)
		

Crossrefs

Compositions contiguously avoiding (1,1) are A003242.

Compositions with some part > 2 are A008466.

Compositions by number of adjacent equal parts are A106356.

Compositions where each part is adjacent to an equal part are A114901.

Compositions contiguously avoiding (1,1,1) are A128695.

Compositions with adjacent parts coprime are A167606.

Compositions contiguously matching (1,1) are A261983.

Compositions with all equal parts contiguous are A274174.

Patterns contiguously matched by compositions are A335457.

Cf. A005251, A011782, A056986, A032020, A131044, A178470, A242882, A335455, A335458.

Programs

Maple

b:= proc(n, t) option remember; `if`(n=0, 1, add(`if`(abs(t)<>j,
       b(n-j, j), `if`(t=-j, 0, b(n-j, -j))), j=1..n))
    end:
a:= n-> ceil(2^(n-1))-b(n, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 06 2020

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],MatchQ[#,{_,x_,x_,x_,_}]&]],{n,0,10}]
(* Second program: *)
b[n_, t_] := b[n, t] = If[n == 0, 1, Sum[If[Abs[t] != j,
     b[n - j, j], If[t == -j, 0, b[n - j, -j]]], {j, 1, n}]];
a[n_] := Ceiling[2^(n-1)] - b[n, 0];
a /@ Range[0, 40] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

Formula

a(n) = A011782(n) - A128695(n). - Alois P. Heinz, Jul 06 2020

Extensions

a(23)-a(35) from Alois P. Heinz, Jul 06 2020