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 A335513 #10 Jun 30 2020 01:55:01
%S A335513 0,1,2,3,4,5,6,8,9,10,11,12,13,14,16,17,18,19,20,21,22,24,25,26,28,32,
%T A335513 33,34,35,36,37,38,40,41,43,44,45,46,48,49,50,52,53,54,56,58,64,65,66,
%U A335513 67,68,69,70,72,73,74,75,76,77,78,80,81,82,83,84,88,89
%N A335513 Numbers k such that the k-th composition in standard order (A066099) avoids the pattern (1,1,1).
%C A335513 These are compositions with no part appearing more than twice.
%C A335513 A composition of n is a finite sequence of positive integers summing to n. 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 A335513 We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).
%H A335513 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>
%H A335513 Gus Wiseman, <a href="https://oeis.org/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>
%H A335513 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>
%e A335513 The sequence of terms together with the corresponding compositions begins:
%e A335513 0: () 17: (4,1) 37: (3,2,1)
%e A335513 1: (1) 18: (3,2) 38: (3,1,2)
%e A335513 2: (2) 19: (3,1,1) 40: (2,4)
%e A335513 3: (1,1) 20: (2,3) 41: (2,3,1)
%e A335513 4: (3) 21: (2,2,1) 43: (2,2,1,1)
%e A335513 5: (2,1) 22: (2,1,2) 44: (2,1,3)
%e A335513 6: (1,2) 24: (1,4) 45: (2,1,2,1)
%e A335513 8: (4) 25: (1,3,1) 46: (2,1,1,2)
%e A335513 9: (3,1) 26: (1,2,2) 48: (1,5)
%e A335513 10: (2,2) 28: (1,1,3) 49: (1,4,1)
%e A335513 11: (2,1,1) 32: (6) 50: (1,3,2)
%e A335513 12: (1,3) 33: (5,1) 52: (1,2,3)
%e A335513 13: (1,2,1) 34: (4,2) 53: (1,2,2,1)
%e A335513 14: (1,1,2) 35: (4,1,1) 54: (1,2,1,2)
%e A335513 16: (5) 36: (3,3) 56: (1,1,4)
%t A335513 stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
%t A335513 Select[Range[0,100],!MatchQ[stc[#],{___,x_,___,x_,___,x_,___}]&]
%Y A335513 These compositions are counted by A232432 (by sum).
%Y A335513 The (1,1)-avoiding version is A233564.
%Y A335513 The complement A335512 is the matching version.
%Y A335513 Constant patterns are counted by A000005 and ranked by A272919.
%Y A335513 Permutations are counted by A000142 and ranked by A333218.
%Y A335513 Patterns are counted by A000670 and ranked by A333217.
%Y A335513 Patterns avoiding (1,1,1) are counted by A080599 (by length).
%Y A335513 Non-unimodal compositions are counted by A115981 and ranked by A335373.
%Y A335513 Combinatory separations are counted by A269134.
%Y A335513 Patterns matched by standard compositions are counted by A335454.
%Y A335513 Minimal patterns avoided by a standard composition are counted by A335465.
%Y A335513 Permutations of prime indices avoiding (1,1,1) are counted by A335511.
%Y A335513 Cf. A034691, A056986, A108917, A114994, A238279, A334968, A335456, A335458.
%K A335513 nonn
%O A335513 1,3
%A A335513 _Gus Wiseman_, Jun 18 2020