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A353508
Number of integer compositions of n with no ones or runs of length 1.
4
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
0,7
LINKS
EXAMPLE
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
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}]
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.
Sequence in context: A368581 A337563 A278648 * A029181 A261426 A260574
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 17 2022
EXTENSIONS
a(41)-a(52) from Alois P. Heinz, May 17 2022
STATUS
approved