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.

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, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 67, 69, 70, 71, 73, 75, 76, 77, 78, 79, 81, 83, 85, 86, 87

Offset: 1

Author

Gus Wiseman, Sep 02 2022

Keywords

• nonn

Comments

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.

Examples

			The terms, binary expansions, and standard compositions:
   1:       1  (1)
   3:      11  (1,1)
   5:     101  (2,1)
   6:     110  (1,2)
   7:     111  (1,1,1)
   9:    1001  (3,1)
  11:    1011  (2,1,1)
  12:    1100  (1,3)
  13:    1101  (1,2,1)
  14:    1110  (1,1,2)
  15:    1111  (1,1,1,1)
  17:   10001  (4,1)
  19:   10011  (3,1,1)
  21:   10101  (2,2,1)
  22:   10110  (2,1,2)
  23:   10111  (2,1,1,1)
  24:   11000  (1,4)
  25:   11001  (1,3,1)
  26:   11010  (1,2,2)
  27:   11011  (1,2,1,1)
  28:   11100  (1,1,3)
  29:   11101  (1,1,2,1)
  30:   11110  (1,1,1,2)
  31:   11111  (1,1,1,1,1)
		

Links

• Gus Wiseman, Statistics, classes, and transformations of standard compositions

Crossrefs

See link for sequences related to standard compositions.

The case beginning with 1 is A004760, complement A004754.

The complement is A022340.

These compositions are counted by A099036, complement A212804.

The case covering an initial interval is A333217.

The gapless but non-initial version is A356843, unordered A356845.

Cf. A004780, A005408, A055932, A073492, A073493, A132747.

Programs

Mathematica

Select[Range[0,100],OddQ[#]||MatchQ[IntegerDigits[#,2],{_,1,1,_}]&]

Formula

Union of A005408 and A004780.