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A353400 Number of integer compositions of n with all run-lengths > 2.

%I A353400 #10 May 17 2022 19:21:17
%S A353400 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 A353400 470,632,914,1198,1611,2286,3018,4079,5709,7619,10329,14333,19258,
%U A353400 26142,36069,48688,66114,90800,122913,167020,228735,310167,421708,576499,782803
%N A353400 Number of integer compositions of n with all run-lengths > 2.
%H A353400 Alois P. Heinz, <a href="/A353400/b353400.txt">Table of n, a(n) for n = 0..1000</a>
%e A353400 The a(7) = 1 through a(12) = 11 compositions:
%e A353400   1111111   2222       333         22222        1112222       444
%e A353400             11111111   111222      1111222      2222111       3333
%e A353400                        222111      2221111      11111222      111333
%e A353400                        111111111   1111111111   22211111      222222
%e A353400                                                 11111111111   333111
%e A353400                                                               11112222
%e A353400                                                               22221111
%e A353400                                                               111111222
%e A353400                                                               111222111
%e A353400                                                               222111111
%e A353400                                                               111111111111
%p A353400 b:= proc(n, h) option remember; `if`(n=0, 1, add(
%p A353400      `if`(i<>h, add(b(n-i*j, i), j=3..n/i), 0), i=1..n/3))
%p A353400     end:
%p A353400 a:= n-> b(n, 0):
%p A353400 seq(a(n), n=0..50);  # _Alois P. Heinz_, May 17 2022
%t A353400 Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n],!MemberQ[Length/@Split[#],1|2]&]],{n,0,15}]
%Y A353400 The = 2 version is A003242 aerated.
%Y A353400 The <= 1 version is A003242 ranked by A333489.
%Y A353400 The version for parts instead of run-lengths is A078012, both A353428.
%Y A353400 The version for partitions is A100405.
%Y A353400 The > 1 version is A114901, ranked by A353427.
%Y A353400 The <= 2 version is A128695, matching A335464.
%Y A353400 A008466 counts compositions with some part > 2.
%Y A353400 A011782 counts compositions.
%Y A353400 A106356 counts compositions by number of adjacent equal parts.
%Y A353400 A274174 counts compositions with equal parts contiguous.
%Y A353400 A329738 counts uniform compositions, partitions A047966.
%Y A353400 A329739 counts compositions with all distinct run-lengths.
%Y A353400 Cf. A005811, A098859, A165413, A175413, A333755, A351013, A353390, A353391, A353401.
%K A353400 nonn
%O A353400 0,7
%A A353400 _Gus Wiseman_, May 15 2022
%E A353400 a(21)-a(49) from _Alois P. Heinz_, May 17 2022