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 A345169 #14 Nov 05 2021 21:54:06
%S A345169 37,52,69,101,104,105,133,137,150,165,180,197,200,208,209,210,261,265,
%T A345169 274,278,300,301,308,325,328,357,360,361,389,393,400,401,406,416,417,
%U A345169 418,421,422,436,517,521,529,530,534,549,550,556,557,564,581,600,601,613
%N A345169 Numbers k such that the k-th composition in standard order is a non-alternating anti-run.
%C A345169 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.
%C A345169 A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).
%C A345169 An anti-run (separation or Carlitz composition) is a sequence with no adjacent equal parts.
%H A345169 Wikipedia, <a href="https://en.wikipedia.org/wiki/Alternating_permutation">Alternating permutation</a>
%F A345169 Intersection of A345168 (non-alternating) and A333489 (anti-run).
%e A345169 The sequence of terms together with their binary indices begins:
%e A345169 37: (3,2,1) 210: (1,2,3,2) 400: (1,3,5)
%e A345169 52: (1,2,3) 261: (6,2,1) 401: (1,3,4,1)
%e A345169 69: (4,2,1) 265: (5,3,1) 406: (1,3,2,1,2)
%e A345169 101: (1,3,2,1) 274: (4,3,2) 416: (1,2,6)
%e A345169 104: (1,2,4) 278: (4,2,1,2) 417: (1,2,5,1)
%e A345169 105: (1,2,3,1) 300: (3,2,1,3) 418: (1,2,4,2)
%e A345169 133: (5,2,1) 301: (3,2,1,2,1) 421: (1,2,3,2,1)
%e A345169 137: (4,3,1) 308: (3,1,2,3) 422: (1,2,3,1,2)
%e A345169 150: (3,2,1,2) 325: (2,4,2,1) 436: (1,2,1,2,3)
%e A345169 165: (2,3,2,1) 328: (2,3,4) 517: (7,2,1)
%e A345169 180: (2,1,2,3) 357: (2,1,3,2,1) 521: (6,3,1)
%e A345169 197: (1,4,2,1) 360: (2,1,2,4) 529: (5,4,1)
%e A345169 200: (1,3,4) 361: (2,1,2,3,1) 530: (5,3,2)
%e A345169 208: (1,2,5) 389: (1,5,2,1) 534: (5,2,1,2)
%e A345169 209: (1,2,4,1) 393: (1,4,3,1) 549: (4,3,2,1)
%t A345169 stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A345169 wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
%t A345169 sepQ[y_]:=!MatchQ[y,{___,x_,x_,___}];
%t A345169 Select[Range[0,1000],sepQ[stc[#]]&&!wigQ[stc[#]]&]
%Y A345169 A version counting partitions is A345166, ranked by A345173.
%Y A345169 These compositions are counted by A345195.
%Y A345169 A001250 counts alternating permutations, complement A348615.
%Y A345169 A003242 counts anti-run compositions.
%Y A345169 A005649 counts anti-run patterns.
%Y A345169 A025047 counts alternating or wiggly compositions, also A025048, A025049.
%Y A345169 A325534 counts separable partitions, ranked by A335433.
%Y A345169 A325535 counts inseparable partitions, ranked by A335448.
%Y A345169 A345164 counts alternating permutations of prime indices.
%Y A345169 A345165 counts partitions w/o an alternating permutation, ranked by A345171.
%Y A345169 A345170 counts partitions w/ an alternating permutation, ranked by A345172.
%Y A345169 A345192 counts non-alternating compositions.
%Y A345169 A345194 counts alternating patterns (with twins: A344605).
%Y A345169 Statistics of standard compositions:
%Y A345169 - Length is A000120.
%Y A345169 - Constant runs are A124767.
%Y A345169 - Heinz number is A333219.
%Y A345169 - Anti-runs are A333381.
%Y A345169 - Runs-resistance is A333628.
%Y A345169 - Number of distinct parts is A334028.
%Y A345169 - Non-anti-runs are A348612.
%Y A345169 Classes of standard compositions:
%Y A345169 - Weakly decreasing compositions (partitions) are A114994.
%Y A345169 - Weakly increasing compositions (multisets) are A225620.
%Y A345169 - Strict compositions are A233564.
%Y A345169 - Constant compositions are A272919.
%Y A345169 - Strictly increasing compositions (sets) are A333255.
%Y A345169 - Strictly decreasing compositions (strict partitions) are A333256.
%Y A345169 - Anti-runs are A333489.
%Y A345169 - Alternating compositions are A345167.
%Y A345169 - Non-Alternating compositions are A345168.
%Y A345169 Cf. A001222, A008965, A238279, A344614, A344615, A344652, A344653, A344654, A345162, A345163, A345193, A348609, A348613.
%K A345169 nonn
%O A345169 1,1
%A A345169 _Gus Wiseman_, Jun 15 2021