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A335488 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,1).

%I A335488 #8 Jun 30 2020 09:55:51
%S A335488 3,7,10,11,13,14,15,19,21,22,23,25,26,27,28,29,30,31,35,36,39,42,43,
%T A335488 45,46,47,49,51,53,54,55,56,57,58,59,60,61,62,63,67,71,73,74,75,76,77,
%U A335488 78,79,82,83,84,85,86,87,89,90,91,92,93,94,95,97,99,100,101
%N A335488 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,1).
%C A335488 These are compositions with some part appearing more than once, or non-strict compositions.
%C A335488 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 A335488 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 A335488 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>
%H A335488 Gus Wiseman, <a href="https://oeis.org/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>
%H A335488 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 A335488 The sequence of terms together with the corresponding compositions begins:
%e A335488    3: (1,1)
%e A335488    7: (1,1,1)
%e A335488   10: (2,2)
%e A335488   11: (2,1,1)
%e A335488   13: (1,2,1)
%e A335488   14: (1,1,2)
%e A335488   15: (1,1,1,1)
%e A335488   19: (3,1,1)
%e A335488   21: (2,2,1)
%e A335488   22: (2,1,2)
%e A335488   23: (2,1,1,1)
%e A335488   25: (1,3,1)
%e A335488   26: (1,2,2)
%e A335488   27: (1,2,1,1)
%e A335488   28: (1,1,3)
%t A335488 stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
%t A335488 Select[Range[0,100],MatchQ[stc[#],{___,x_,___,x_,___}]&]
%Y A335488 The complement A233564 is the avoiding version.
%Y A335488 Patterns matching this pattern are counted by A019472 (by length).
%Y A335488 Permutations of prime indices matching this pattern are counted by A335487.
%Y A335488 These compositions are counted by A261982 (by sum).
%Y A335488 Constant patterns are counted by A000005 and ranked by A272919.
%Y A335488 Permutations are counted by A000142 and ranked by A333218.
%Y A335488 Patterns are counted by A000670 and ranked by A333217.
%Y A335488 Non-unimodal compositions are counted by A115981 and ranked by A335373.
%Y A335488 Combinatory separations are counted by A269134.
%Y A335488 Patterns matched by standard compositions are counted by A335454.
%Y A335488 Minimal patterns avoided by a standard composition are counted by A335465.
%Y A335488 The (1,1,1)-matching case is A335512.
%Y A335488 Cf. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A334968, A335456, A335458.
%K A335488 nonn
%O A335488 1,1
%A A335488 _Gus Wiseman_, Jun 18 2020