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%I #10 Jun 12 2021 06:05:07
%S 1,1,2,3,6,10,17,29,50,84,143,241,408,688,1162,1959,3305,5571,9393,
%T 15832,26688,44980,75812,127769,215338,362911,611620,1030758,1737131,
%U 2927556,4933760,8314754,14012668,23615198,39798098,67070686,113032453,190490542,321028554
%N Number of compositions of n with no adjacent triples (..., x, y, z, ...) where x <= y <= z.
%C These compositions avoid the weak consecutive pattern (1,2,3), the strict version being A128761.
%e The a(1) = 1 through a(6) = 17 compositions:
%e (1) (2) (3) (4) (5) (6)
%e (1,1) (1,2) (1,3) (1,4) (1,5)
%e (2,1) (2,2) (2,3) (2,4)
%e (3,1) (3,2) (3,3)
%e (1,2,1) (4,1) (4,2)
%e (2,1,1) (1,3,1) (5,1)
%e (2,1,2) (1,3,2)
%e (2,2,1) (1,4,1)
%e (3,1,1) (2,1,3)
%e (1,2,1,1) (2,3,1)
%e (3,1,2)
%e (3,2,1)
%e (4,1,1)
%e (1,2,1,2)
%e (1,3,1,1)
%e (2,1,2,1)
%e (2,2,1,1)
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{___,x_,y_,z_,___}/;x<=y<=z]&]],{n,0,15}]
%Y The case of permutations is A049774.
%Y The strict non-adjacent version is A102726.
%Y The case of permutations of prime indices is A344652.
%Y A001250 counts alternating permutations.
%Y A005649 counts anti-run patterns.
%Y A106356 counts compositions by number of maximal anti-runs.
%Y A114901 counts compositions where each part is adjacent to an equal part.
%Y A344604 counts wiggly compositions with twins.
%Y A344605 counts wiggly patterns with twins.
%Y A344606 counts wiggly permutations of prime factors with twins.
%Y Counting compositions by patterns:
%Y - A003242 avoiding (1,1) adjacent.
%Y - A011782 no conditions.
%Y - A106351 avoiding (1,1) adjacent by sum and length.
%Y - A128695 avoiding (1,1,1) adjacent.
%Y - A128761 avoiding (1,2,3).
%Y - A232432 avoiding (1,1,1).
%Y - A335456 all patterns.
%Y - A335457 all patterns adjacent.
%Y - A335514 matching (1,2,3).
%Y - A344604 weakly avoiding (1,2,3) and (3,2,1) adjacent.
%Y - A344614 avoiding (1,2,3) and (3,2,1) adjacent.
%Y - A344615 weakly avoiding (1,2,3) adjacent.
%Y Cf. A000041, A006330, A008965, A027187, A238279/A333755, A333213, A335464, A335515, A344612, A344619.
%K nonn
%O 0,3
%A _Gus Wiseman_, May 27 2021
%E More terms from _Bert Dobbelaere_, Jun 12 2021
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