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A353428 Number of integer compositions of n with all parts and all run-lengths > 2.

%I A353428 #12 May 18 2022 13:33:42
%S A353428 1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,2,1,0,2,0,2,4,0,0,8,3,0,10,4,4,15,4,8,
%T A353428 24,7,8,42,16,10,59,31,27,87,37,52,149,62,66,233,121,111,342,207,204,
%U A353428 531,308,351,864,487,536,1373,864,865,2057,1440,1509,3232
%N A353428 Number of integer compositions of n with all parts and all run-lengths > 2.
%H A353428 Alois P. Heinz, <a href="/A353428/b353428.txt">Table of n, a(n) for n = 0..2000</a>
%e A353428 The a(n) compositions for selected n:
%e A353428   n=16:   n=18:     n=20:    n=21:      n=24:
%e A353428 ----------------------------------------------------
%e A353428   (4444)  (666)     (5555)   (777)      (888)
%e A353428           (333333)  (44444)  (333444)   (6666)
%e A353428                              (444333)   (333555)
%e A353428                              (3333333)  (444444)
%e A353428                                         (555333)
%e A353428                                         (3333444)
%e A353428                                         (4443333)
%e A353428                                         (33333333)
%p A353428 b:= proc(n, h) option remember; `if`(n=0, 1, add(
%p A353428      `if`(i=h, 0, add(b(n-i*j, i), j=3..n/i)), i=3..n/3))
%p A353428     end:
%p A353428 a:= n-> b(n, 0):
%p A353428 seq(a(n), n=0..80);  # _Alois P. Heinz_, May 18 2022
%t A353428 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[#,1|2]&&!MemberQ[Length/@Split[#],1|2]&]],{n,0,15}]
%Y A353428 Allowing any multiplicities gives A078012, partitions A008483.
%Y A353428 The version for no (instead of all) parts or run-lengths > 2 is A137200.
%Y A353428 Allowing any parts gives A353400, partitions A100405.
%Y A353428 The version for partitions is A353501, ranked by A353502.
%Y A353428 The version for > 1 instead of > 2 is A353508, partitions A339222.
%Y A353428 A003242 counts anti-run compositions, ranked by A333489.
%Y A353428 A008466 counts compositions with some part > 2.
%Y A353428 A011782 counts compositions.
%Y A353428 A114901 counts compositions with no runs of length 1, ranked by A353427.
%Y A353428 A128695 counts compositions with no run-lengths > 2.
%Y A353428 A261983 counts non-anti-run compositions.
%Y A353428 A335464 counts compositions with a run-length > 2.
%Y A353428 Cf. A032020, A044813, A098859, A165413, A318928, A329738, A329739, A333755, A353390, A353429.
%K A353428 nonn
%O A353428 0,13
%A A353428 _Gus Wiseman_, May 16 2022
%E A353428 a(26)-a(66) from _Alois P. Heinz_, May 17 2022