This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A060142 #50 Jan 05 2025 19:51:36
%S A060142 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,
%T A060142 79,97,100,103,112,115,121,124,127,129,132,135,144,147,153,156,159,
%U A060142 192,195,201,204,207,225,228,231,240,243,249,252,255,256,259,265,268,271
%N 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.
%C A060142 After expelling 0 and 1, the numbers 4x occupy same positions in S that 1 occupies in the infinite Fibonacci word (A003849).
%C A060142 a(A026351(n)) = A219608(n); a(A004957(n)) = 4 * a(n). - _Reinhard Zumkeller_, Nov 26 2012
%C A060142 Apart from the initial term, this lists the indices of the 1's in A086747. - _N. J. A. Sloane_, Dec 05 2019
%C A060142 From _Gus Wiseman_, Jun 10 2020: (Start)
%C A060142 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:
%C A060142 0: () 57: (1,1,3,1) 135: (5,1,1,1)
%C A060142 1: (1) 60: (1,1,1,3) 144: (3,5)
%C A060142 3: (1,1) 63: (1,1,1,1,1,1) 147: (3,3,1,1)
%C A060142 4: (3) 64: (7) 153: (3,1,3,1)
%C A060142 7: (1,1,1) 67: (5,1,1) 156: (3,1,1,3)
%C A060142 9: (3,1) 73: (3,3,1) 159: (3,1,1,1,1,1)
%C A060142 12: (1,3) 76: (3,1,3) 192: (1,7)
%C A060142 15: (1,1,1,1) 79: (3,1,1,1,1) 195: (1,5,1,1)
%C A060142 16: (5) 97: (1,5,1) 201: (1,3,3,1)
%C A060142 19: (3,1,1) 100: (1,3,3) 204: (1,3,1,3)
%C A060142 25: (1,3,1) 103: (1,3,1,1,1) 207: (1,3,1,1,1,1)
%C A060142 28: (1,1,3) 112: (1,1,5) 225: (1,1,5,1)
%C A060142 31: (1,1,1,1,1) 115: (1,1,3,1,1) 228: (1,1,3,3)
%C A060142 33: (5,1) 121: (1,1,1,3,1) 231: (1,1,3,1,1,1)
%C A060142 36: (3,3) 124: (1,1,1,1,3) 240: (1,1,1,5)
%C A060142 39: (3,1,1,1) 127: (1,1,1,1,1,1,1) 243: (1,1,1,3,1,1)
%C A060142 48: (1,5) 129: (7,1) 249: (1,1,1,1,3,1)
%C A060142 51: (1,3,1,1) 132: (5,3) 252: (1,1,1,1,1,3)
%C A060142 (End)
%C A060142 Numbers whose binary representation has the property that every run of consecutive 0's has even length. - _Harry Richman_, Jan 31 2024
%H A060142 Reinhard Zumkeller, <a href="/A060142/b060142.txt">Table of n, a(n) for n = 0..10000</a>
%H A060142 Clark Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/goldentext.html">A Self-Generating Set and the Golden Mean</a>, J. Integer Sequences, 3 (2000), #00.2.8.
%H A060142 Clark Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/52-3/Kimberling11132013.pdf">Fusion, Fission, and Factors</a>, Fib. Q., 52 (2014), 195-202.
%H A060142 Lukasz Merta, <a href="https://arxiv.org/abs/1803.00292">Composition inverses of the variations of the Baum-Sweet sequence</a>, arXiv:1803.00292 [math.NT], 2018. See l(n) p. 5.
%H A060142 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>
%e A060142 From _Harry Richman_, Jan 31 2024: (Start)
%e A060142 In the following, dots are used for zeros in the binary representation:
%e A060142 n binary(a(n)) a(n)
%e A060142 0: ....... 0
%e A060142 1: ......1 1
%e A060142 2: .....11 3
%e A060142 3: ....1.. 4
%e A060142 4: ....111 7
%e A060142 5: ...1..1 9
%e A060142 6: ...11.. 12
%e A060142 7: ...1111 15
%e A060142 8: ..1.... 16
%e A060142 9: ..1..11 19
%e A060142 10: ..11..1 25
%e A060142 11: ..111.. 28
%e A060142 12: ..11111 31
%e A060142 13: .1....1 33
%e A060142 14: .1..1.. 36
%e A060142 15: .1..111 39
%e A060142 16: .11.... 48
%e A060142 17: .11..11 51
%e A060142 18: .111..1 57
%e A060142 19: .1111.. 60
%e A060142 20: .111111 63
%e A060142 21: 1...... 64
%e A060142 22: 1....11 67
%e A060142 (End)
%t A060142 Take[Nest[Union[Flatten[# /. {{i_Integer -> i}, {i_Integer -> 2 i + 1}, {i_Integer -> 4 i}}]] &, {1}, 5], 32] (* Or *)
%t A060142 Select[Range[124], FreeQ[Length /@ Select[Split[IntegerDigits[#, 2]], First[#] == 0 &], _?OddQ] &] (* _Birkas Gyorgy_, May 29 2012 *)
%o A060142 (Haskell)
%o A060142 import Data.Set (singleton, deleteFindMin, insert)
%o A060142 a060142 n = a060142_list !! n
%o A060142 a060142_list = 0 : f (singleton 1) where
%o A060142 f s = x : f (insert (4 * x) $ insert (2 * x + 1) s') where
%o A060142 (x, s') = deleteFindMin s
%o A060142 -- _Reinhard Zumkeller_, Nov 26 2012
%o A060142 (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
%Y A060142 Cf. A219608 (odd terms), A060138, A060139, A060140, A060141, A086747.
%Y A060142 Cf. A003714 (no consecutive 1's in binary expansion).
%Y A060142 Odd partitions are counted by A000009.
%Y A060142 Numbers with an odd number of 1's in binary expansion are A000069.
%Y A060142 Numbers whose binary expansion has odd length are A053738.
%Y A060142 All of the following pertain to compositions in standard order (A066099):
%Y A060142 - Length is A000120.
%Y A060142 - Compositions without odd parts are A062880.
%Y A060142 - Sum is A070939.
%Y A060142 - Product is A124758.
%Y A060142 - Strict compositions are A233564.
%Y A060142 - Heinz number is A333219.
%Y A060142 - Number of distinct parts is A334028.
%K A060142 nonn
%O A060142 0,3
%A A060142 _Clark Kimberling_, Mar 05 2001
%E A060142 Corrected by _T. D. Noe_, Nov 01 2006
%E A060142 Definition simplified by _Charles R Greathouse IV_, Oct 21 2013