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 A337459 #12 Feb 16 2025 08:34:00
%S A337459 7,11,13,14,19,21,25,26,28,35,37,41,42,49,50,52,56,67,69,73,74,81,82,
%T A337459 84,97,98,100,104,112,131,133,137,138,145,146,161,162,164,168,193,194,
%U A337459 196,200,208,224,259,261,265,266,273,274,289,290,292,321,322,324
%N A337459 Numbers k such that the k-th composition in standard order is a unimodal triple.
%C A337459 A composition of n is a finite sequence of positive integers summing to n.
%C A337459 A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%C A337459 The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
%H A337459 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%F A337459 Complement of A335373 in A014311.
%e A337459 The sequence together with the corresponding triples begins:
%e A337459 7: (1,1,1) 52: (1,2,3) 133: (5,2,1)
%e A337459 11: (2,1,1) 56: (1,1,4) 137: (4,3,1)
%e A337459 13: (1,2,1) 67: (5,1,1) 138: (4,2,2)
%e A337459 14: (1,1,2) 69: (4,2,1) 145: (3,4,1)
%e A337459 19: (3,1,1) 73: (3,3,1) 146: (3,3,2)
%e A337459 21: (2,2,1) 74: (3,2,2) 161: (2,5,1)
%e A337459 25: (1,3,1) 81: (2,4,1) 162: (2,4,2)
%e A337459 26: (1,2,2) 82: (2,3,2) 164: (2,3,3)
%e A337459 28: (1,1,3) 84: (2,2,3) 168: (2,2,4)
%e A337459 35: (4,1,1) 97: (1,5,1) 193: (1,6,1)
%e A337459 37: (3,2,1) 98: (1,4,2) 194: (1,5,2)
%e A337459 41: (2,3,1) 100: (1,3,3) 196: (1,4,3)
%e A337459 42: (2,2,2) 104: (1,2,4) 200: (1,3,4)
%e A337459 49: (1,4,1) 112: (1,1,5) 208: (1,2,5)
%e A337459 50: (1,3,2) 131: (6,1,1) 224: (1,1,6)
%t A337459 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A337459 Select[Range[0,1000],Length[stc[#]]==3&&!MatchQ[stc[#],{x_,y_,z_}/;x>y<z]&]
%Y A337459 A337460 is the non-unimodal version.
%Y A337459 A000217(n - 2) counts 3-part compositions.
%Y A337459 6*A001399(n - 6) = 6*A069905(n - 3) = 6*A211540(n - 1) counts strict 3-part compositions.
%Y A337459 A001399(n - 3) = A069905(n) = A211540(n + 2) counts 3-part partitions.
%Y A337459 A001399(n - 6) = A069905(n - 3) = A211540(n - 1) counts strict 3-part partitions.
%Y A337459 A001523 counts unimodal compositions.
%Y A337459 A007052 counts unimodal patterns.
%Y A337459 A011782 counts unimodal permutations.
%Y A337459 A115981 counts non-unimodal compositions.
%Y A337459 All of the following pertain to compositions in standard order (A066099):
%Y A337459 - Length is A000120.
%Y A337459 - Triples are A014311, with strict case A337453.
%Y A337459 - Sum is A070939.
%Y A337459 - Runs are counted by A124767.
%Y A337459 - Strict compositions are A233564.
%Y A337459 - Constant compositions are A272919.
%Y A337459 - Heinz number is A333219.
%Y A337459 - Combinatory separations are counted by A334030.
%Y A337459 - Non-unimodal compositions are A335373.
%Y A337459 - Non-co-unimodal compositions are A335374.
%Y A337459 Cf. A007304, A014612, A072706, A156040, A211540, A227038, A332743, A337452, A337461, A337604.
%K A337459 nonn
%O A337459 1,1
%A A337459 _Gus Wiseman_, Sep 07 2020