A060142 Ordered set S defined by these rules: 0 is in S and if x is in S then 2x+1 and 4x are in S.

0, 1, 3, 4, 7, 9, 12, 15, 16, 19, 25, 28, 31, 33, 36, 39, 48, 51, 57, 60, 63, 64, 67, 73, 76, 79, 97, 100, 103, 112, 115, 121, 124, 127, 129, 132, 135, 144, 147, 153, 156, 159, 192, 195, 201, 204, 207, 225, 228, 231, 240, 243, 249, 252, 255, 256, 259, 265, 268, 271

Offset: 0

Author

Clark Kimberling, Mar 05 2001

Keywords

• nonn

Comments

After expelling 0 and 1, the numbers 4x occupy same positions in S that 1 occupies in the infinite Fibonacci word (A003849).

a(A026351(n)) = A219608(n); a(A004957(n)) = 4 * a(n). - Reinhard Zumkeller, Nov 26 2012

Apart from the initial term, this lists the indices of the 1's in A086747. - N. J. A. Sloane, Dec 05 2019

From Gus Wiseman, Jun 10 2020: (Start)

Numbers k such that the k-th composition in standard order has all odd parts, or numbers k such that A124758(k) is odd. 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. For example, the sequence of all compositions into odd parts begins:

0: () 57: (1,1,3,1) 135: (5,1,1,1)

1: (1) 60: (1,1,1,3) 144: (3,5)

3: (1,1) 63: (1,1,1,1,1,1) 147: (3,3,1,1)

4: (3) 64: (7) 153: (3,1,3,1)

7: (1,1,1) 67: (5,1,1) 156: (3,1,1,3)

9: (3,1) 73: (3,3,1) 159: (3,1,1,1,1,1)

12: (1,3) 76: (3,1,3) 192: (1,7)

15: (1,1,1,1) 79: (3,1,1,1,1) 195: (1,5,1,1)

16: (5) 97: (1,5,1) 201: (1,3,3,1)

19: (3,1,1) 100: (1,3,3) 204: (1,3,1,3)

25: (1,3,1) 103: (1,3,1,1,1) 207: (1,3,1,1,1,1)

28: (1,1,3) 112: (1,1,5) 225: (1,1,5,1)

31: (1,1,1,1,1) 115: (1,1,3,1,1) 228: (1,1,3,3)

33: (5,1) 121: (1,1,1,3,1) 231: (1,1,3,1,1,1)

36: (3,3) 124: (1,1,1,1,3) 240: (1,1,1,5)

39: (3,1,1,1) 127: (1,1,1,1,1,1,1) 243: (1,1,1,3,1,1)

48: (1,5) 129: (7,1) 249: (1,1,1,1,3,1)

51: (1,3,1,1) 132: (5,3) 252: (1,1,1,1,1,3)

(End)

Numbers whose binary representation has the property that every run of consecutive 0's has even length. - Harry Richman, Jan 31 2024

Examples

			From _Harry Richman_, Jan 31 2024: (Start)
In the following, dots are used for zeros in the binary representation:
   n  binary(a(n))  a(n)
   0:    .......     0
   1:    ......1     1
   2:    .....11     3
   3:    ....1..     4
   4:    ....111     7
   5:    ...1..1     9
   6:    ...11..    12
   7:    ...1111    15
   8:    ..1....    16
   9:    ..1..11    19
  10:    ..11..1    25
  11:    ..111..    28
  12:    ..11111    31
  13:    .1....1    33
  14:    .1..1..    36
  15:    .1..111    39
  16:    .11....    48
  17:    .11..11    51
  18:    .111..1    57
  19:    .1111..    60
  20:    .111111    63
  21:    1......    64
  22:    1....11    67
(End)
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Clark Kimberling, A Self-Generating Set and the Golden Mean, J. Integer Sequences, 3 (2000), #00.2.8.
• Clark Kimberling, Fusion, Fission, and Factors, Fib. Q., 52 (2014), 195-202.
• Lukasz Merta, Composition inverses of the variations of the Baum-Sweet sequence, arXiv:1803.00292 [math.NT], 2018. See l(n) p. 5.
• Gus Wiseman, Statistics, classes, and transformations of standard compositions

Crossrefs

Cf. A219608 (odd terms), A060138, A060139, A060140, A060141, A086747.

Cf. A003714 (no consecutive 1's in binary expansion).

Odd partitions are counted by A000009.

Numbers with an odd number of 1's in binary expansion are A000069.

Numbers whose binary expansion has odd length are A053738.

All of the following pertain to compositions in standard order (A066099):

- Length is A000120.

- Compositions without odd parts are A062880.

- Sum is A070939.

- Product is A124758.

- Strict compositions are A233564.

- Heinz number is A333219.

- Number of distinct parts is A334028.

Programs

Haskell

import Data.Set (singleton, deleteFindMin, insert)
a060142 n = a060142_list !! n
a060142_list = 0 : f (singleton 1) where
   f s = x : f (insert (4 * x) $ insert (2 * x + 1) s') where
       (x, s') = deleteFindMin s
-- Reinhard Zumkeller, Nov 26 2012

Mathematica

Take[Nest[Union[Flatten[# /. {{i_Integer -> i}, {i_Integer -> 2 i + 1}, {i_Integer -> 4 i}}]] &, {1}, 5], 32]  (* Or *)
Select[Range[124], FreeQ[Length /@ Select[Split[IntegerDigits[#, 2]], First[#] == 0 &], ?OddQ] &] (* _Birkas Gyorgy, May 29 2012 *)

PARI

is(n)=if(n<3, n<2, if(n%2,is(n\2),n%4==0 && is(n/4))) \\ Charles R Greathouse IV, Oct 21 2013

Extensions

Corrected by T. D. Noe, Nov 01 2006

Definition simplified by Charles R Greathouse IV, Oct 21 2013