login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 15
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 (list; graph; refs; listen; history; text; internal format)
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).

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)

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

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

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.

Sequence in context: A185256 A070992 A246514 * A049844 A248358 A244952

Adjacent sequences:  A060139 A060140 A060141 * A060143 A060144 A060145

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 23 12:02 EDT 2021. Contains 347613 sequences. (Running on oeis4.)