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Number of integer compositions of n with all run-lengths > 2.
10

%I #10 May 17 2022 19:21:17

%S 1,0,0,1,1,1,2,1,2,4,4,5,11,11,14,27,29,37,61,72,97,147,181,246,368,

%T 470,632,914,1198,1611,2286,3018,4079,5709,7619,10329,14333,19258,

%U 26142,36069,48688,66114,90800,122913,167020,228735,310167,421708,576499,782803

%N Number of integer compositions of n with all run-lengths > 2.

%H Alois P. Heinz, <a href="/A353400/b353400.txt">Table of n, a(n) for n = 0..1000</a>

%e The a(7) = 1 through a(12) = 11 compositions:

%e 1111111 2222 333 22222 1112222 444

%e 11111111 111222 1111222 2222111 3333

%e 222111 2221111 11111222 111333

%e 111111111 1111111111 22211111 222222

%e 11111111111 333111

%e 11112222

%e 22221111

%e 111111222

%e 111222111

%e 222111111

%e 111111111111

%p b:= proc(n, h) option remember; `if`(n=0, 1, add(

%p `if`(i<>h, add(b(n-i*j, i), j=3..n/i), 0), i=1..n/3))

%p end:

%p a:= n-> b(n, 0):

%p seq(a(n), n=0..50); # _Alois P. Heinz_, May 17 2022

%t Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n],!MemberQ[Length/@Split[#],1|2]&]],{n,0,15}]

%Y The = 2 version is A003242 aerated.

%Y The <= 1 version is A003242 ranked by A333489.

%Y The version for parts instead of run-lengths is A078012, both A353428.

%Y The version for partitions is A100405.

%Y The > 1 version is A114901, ranked by A353427.

%Y The <= 2 version is A128695, matching A335464.

%Y A008466 counts compositions with some part > 2.

%Y A011782 counts compositions.

%Y A106356 counts compositions by number of adjacent equal parts.

%Y A274174 counts compositions with equal parts contiguous.

%Y A329738 counts uniform compositions, partitions A047966.

%Y A329739 counts compositions with all distinct run-lengths.

%Y Cf. A005811, A098859, A165413, A175413, A333755, A351013, A353390, A353391, A353401.

%K nonn

%O 0,7

%A _Gus Wiseman_, May 15 2022

%E a(21)-a(49) from _Alois P. Heinz_, May 17 2022