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A356844 Numbers k such that the k-th composition in standard order contains at least one 1. Numbers that are odd or whose binary expansion contains at least two adjacent 1's.

%I A356844 #9 Sep 03 2022 12:20:22
%S A356844 1,3,5,6,7,9,11,12,13,14,15,17,19,21,22,23,24,25,26,27,28,29,30,31,33,
%T A356844 35,37,38,39,41,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,
%U A356844 61,62,63,65,67,69,70,71,73,75,76,77,78,79,81,83,85,86,87
%N A356844 Numbers k such that the k-th composition in standard order contains at least one 1. Numbers that are odd or whose binary expansion contains at least two adjacent 1's.
%C A356844 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 A356844 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>
%F A356844 Union of A005408 and A004780.
%e A356844 The terms, binary expansions, and standard compositions:
%e A356844    1:       1  (1)
%e A356844    3:      11  (1,1)
%e A356844    5:     101  (2,1)
%e A356844    6:     110  (1,2)
%e A356844    7:     111  (1,1,1)
%e A356844    9:    1001  (3,1)
%e A356844   11:    1011  (2,1,1)
%e A356844   12:    1100  (1,3)
%e A356844   13:    1101  (1,2,1)
%e A356844   14:    1110  (1,1,2)
%e A356844   15:    1111  (1,1,1,1)
%e A356844   17:   10001  (4,1)
%e A356844   19:   10011  (3,1,1)
%e A356844   21:   10101  (2,2,1)
%e A356844   22:   10110  (2,1,2)
%e A356844   23:   10111  (2,1,1,1)
%e A356844   24:   11000  (1,4)
%e A356844   25:   11001  (1,3,1)
%e A356844   26:   11010  (1,2,2)
%e A356844   27:   11011  (1,2,1,1)
%e A356844   28:   11100  (1,1,3)
%e A356844   29:   11101  (1,1,2,1)
%e A356844   30:   11110  (1,1,1,2)
%e A356844   31:   11111  (1,1,1,1,1)
%t A356844 Select[Range[0,100],OddQ[#]||MatchQ[IntegerDigits[#,2],{___,1,1,___}]&]
%Y A356844 See link for sequences related to standard compositions.
%Y A356844 The case beginning with 1 is A004760, complement A004754.
%Y A356844 The complement is A022340.
%Y A356844 These compositions are counted by A099036, complement A212804.
%Y A356844 The case covering an initial interval is A333217.
%Y A356844 The gapless but non-initial version is A356843, unordered A356845.
%Y A356844 Cf. A004780, A005408, A055932, A073492, A073493, A132747.
%K A356844 nonn
%O A356844 1,2
%A A356844 _Gus Wiseman_, Sep 02 2022