This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A344615 #10 Jun 12 2021 06:05:07 %S A344615 1,1,2,3,6,10,17,29,50,84,143,241,408,688,1162,1959,3305,5571,9393, %T A344615 15832,26688,44980,75812,127769,215338,362911,611620,1030758,1737131, %U A344615 2927556,4933760,8314754,14012668,23615198,39798098,67070686,113032453,190490542,321028554 %N A344615 Number of compositions of n with no adjacent triples (..., x, y, z, ...) where x <= y <= z. %C A344615 These compositions avoid the weak consecutive pattern (1,2,3), the strict version being A128761. %e A344615 The a(1) = 1 through a(6) = 17 compositions: %e A344615 (1) (2) (3) (4) (5) (6) %e A344615 (1,1) (1,2) (1,3) (1,4) (1,5) %e A344615 (2,1) (2,2) (2,3) (2,4) %e A344615 (3,1) (3,2) (3,3) %e A344615 (1,2,1) (4,1) (4,2) %e A344615 (2,1,1) (1,3,1) (5,1) %e A344615 (2,1,2) (1,3,2) %e A344615 (2,2,1) (1,4,1) %e A344615 (3,1,1) (2,1,3) %e A344615 (1,2,1,1) (2,3,1) %e A344615 (3,1,2) %e A344615 (3,2,1) %e A344615 (4,1,1) %e A344615 (1,2,1,2) %e A344615 (1,3,1,1) %e A344615 (2,1,2,1) %e A344615 (2,2,1,1) %t A344615 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{___,x_,y_,z_,___}/;x<=y<=z]&]],{n,0,15}] %Y A344615 The case of permutations is A049774. %Y A344615 The strict non-adjacent version is A102726. %Y A344615 The case of permutations of prime indices is A344652. %Y A344615 A001250 counts alternating permutations. %Y A344615 A005649 counts anti-run patterns. %Y A344615 A106356 counts compositions by number of maximal anti-runs. %Y A344615 A114901 counts compositions where each part is adjacent to an equal part. %Y A344615 A344604 counts wiggly compositions with twins. %Y A344615 A344605 counts wiggly patterns with twins. %Y A344615 A344606 counts wiggly permutations of prime factors with twins. %Y A344615 Counting compositions by patterns: %Y A344615 - A003242 avoiding (1,1) adjacent. %Y A344615 - A011782 no conditions. %Y A344615 - A106351 avoiding (1,1) adjacent by sum and length. %Y A344615 - A128695 avoiding (1,1,1) adjacent. %Y A344615 - A128761 avoiding (1,2,3). %Y A344615 - A232432 avoiding (1,1,1). %Y A344615 - A335456 all patterns. %Y A344615 - A335457 all patterns adjacent. %Y A344615 - A335514 matching (1,2,3). %Y A344615 - A344604 weakly avoiding (1,2,3) and (3,2,1) adjacent. %Y A344615 - A344614 avoiding (1,2,3) and (3,2,1) adjacent. %Y A344615 - A344615 weakly avoiding (1,2,3) adjacent. %Y A344615 Cf. A000041, A006330, A008965, A027187, A238279/A333755, A333213, A335464, A335515, A344612, A344619. %K A344615 nonn %O A344615 0,3 %A A344615 _Gus Wiseman_, May 27 2021 %E A344615 More terms from _Bert Dobbelaere_, Jun 12 2021