A353508 Number of integer compositions of n with no ones or runs of length 1.

1, 0, 0, 0, 1, 0, 2, 0, 2, 1, 4, 0, 8, 2, 11, 4, 21, 5, 37, 12, 57, 25, 104, 38, 177, 79, 292, 149, 513, 251, 876, 482, 1478, 871, 2562, 1533, 4387, 2815, 7473, 5036, 12908, 8935, 22135, 16085, 37940, 28611, 65422, 50731, 112459, 90408, 193386, 160119, 333513

Offset: 0

Author

Gus Wiseman, May 17 2022

Keywords

• nonn

Examples

			The a(0) = 1 through a(14) = 11 compositions (empty columns indicated by dots, 0 is the empty composition):
  0  .  .  .  22  .  33   .  44    333  55     .  66      22333  77
                     222     2222       2233      444     33322  2255
                                        3322      2244           3344
                                        22222     3333           4433
                                                  4422           5522
                                                  22233          22244
                                                  33222          44222
                                                  222222         222233
                                                                 223322
                                                                 332222
                                                                 2222222
		

Links

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

Crossrefs

The version for partitions is A339222.

Compositions counted by their run-lengths:

- For run-lengths <= 1 we have A003242, ranked by A333489.

- For run-lengths = 2 we have A003242 aerated.

- For run-lengths > 1 we have A114901, ranked by A353427.

- For run-lengths <= 2 we have A128695 matching A335464.

- For run-lengths > 2 we have A353400, partitions A100405.

- For run-lengths all prime we have A353401.

- For run-lengths and parts > 2 we have A353428.

A008466 counts compositions with some part > 2.

A011782 counts compositions.

A106356 counts compositions by number of adjacent equal parts.

A261983 counts non-anti-run compositions.

A274174 counts compositions with equal parts contiguous.

Cf. A005811, A078012, A238279, A329739, A333755, A351013, A353390.

Programs

Maple

b:= proc(n,h) option remember; `if`(n=0, 1, add(
     `if`(i<>h, add(b(n-i*j, i), j=2..n/i), 0), i=2..n/2))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, May 17 2022

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[#,1]&&!MemberQ[Length/@Split[#],1]&]],{n,0,15}]

Extensions

a(41)-a(52) from Alois P. Heinz, May 17 2022