OFFSET
0,3
COMMENTS
After expelling 0 and 1, the numbers 4x occupy same positions in S that 1 occupies in the infinite Fibonacci word (A003849).
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
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.
EXAMPLE
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)
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 *)
PROG
(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
(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
CROSSREFS
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.
KEYWORD
nonn
AUTHOR
Clark Kimberling, Mar 05 2001
EXTENSIONS
Corrected by T. D. Noe, Nov 01 2006
Definition simplified by Charles R Greathouse IV, Oct 21 2013
STATUS
approved