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%I #49 Feb 04 2024 18:44:53
%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,
%T 79,97,100,103,112,115,121,124,127,129,132,135,144,147,153,156,159,
%U 192,195,201,204,207,225,228,231,240,243,249,252,255,256,259,265,268,271
%N 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 After expelling 0 and 1, the numbers 4x occupy same positions in S that 1 occupies in the infinite Fibonacci word (A003849).
%C a(A026351(n)) = A219608(n); a(A004957(n)) = 4 * a(n). - _Reinhard Zumkeller_, Nov 26 2012
%C Apart from the initial term, this lists the indices of the 1's in A086747. - _N. J. A. Sloane_, Dec 05 2019
%C From _Gus Wiseman_, Jun 10 2020: (Start)
%C 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 0: () 57: (1,1,3,1) 135: (5,1,1,1)
%C 1: (1) 60: (1,1,1,3) 144: (3,5)
%C 3: (1,1) 63: (1,1,1,1,1,1) 147: (3,3,1,1)
%C 4: (3) 64: (7) 153: (3,1,3,1)
%C 7: (1,1,1) 67: (5,1,1) 156: (3,1,1,3)
%C 9: (3,1) 73: (3,3,1) 159: (3,1,1,1,1,1)
%C 12: (1,3) 76: (3,1,3) 192: (1,7)
%C 15: (1,1,1,1) 79: (3,1,1,1,1) 195: (1,5,1,1)
%C 16: (5) 97: (1,5,1) 201: (1,3,3,1)
%C 19: (3,1,1) 100: (1,3,3) 204: (1,3,1,3)
%C 25: (1,3,1) 103: (1,3,1,1,1) 207: (1,3,1,1,1,1)
%C 28: (1,1,3) 112: (1,1,5) 225: (1,1,5,1)
%C 31: (1,1,1,1,1) 115: (1,1,3,1,1) 228: (1,1,3,3)
%C 33: (5,1) 121: (1,1,1,3,1) 231: (1,1,3,1,1,1)
%C 36: (3,3) 124: (1,1,1,1,3) 240: (1,1,1,5)
%C 39: (3,1,1,1) 127: (1,1,1,1,1,1,1) 243: (1,1,1,3,1,1)
%C 48: (1,5) 129: (7,1) 249: (1,1,1,1,3,1)
%C 51: (1,3,1,1) 132: (5,3) 252: (1,1,1,1,1,3)
%C (End)
%C Numbers whose binary representation has the property that every run of consecutive 0's has even length. - _Harry Richman_, Jan 31 2024
%H Reinhard Zumkeller, <a href="/A060142/b060142.txt">Table of n, a(n) for n = 0..10000</a>
%H 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 Clark Kimberling, <a href="https://www.fq.math.ca/Papers1/52-3/Kimberling11132013.pdf">Fusion, Fission, and Factors</a>, Fib. Q., 52 (2014), 195-202.
%H 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 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 From _Harry Richman_, Jan 31 2024: (Start)
%e In the following, dots are used for zeros in the binary representation:
%e n binary(a(n)) a(n)
%e 0: ....... 0
%e 1: ......1 1
%e 2: .....11 3
%e 3: ....1.. 4
%e 4: ....111 7
%e 5: ...1..1 9
%e 6: ...11.. 12
%e 7: ...1111 15
%e 8: ..1.... 16
%e 9: ..1..11 19
%e 10: ..11..1 25
%e 11: ..111.. 28
%e 12: ..11111 31
%e 13: .1....1 33
%e 14: .1..1.. 36
%e 15: .1..111 39
%e 16: .11.... 48
%e 17: .11..11 51
%e 18: .111..1 57
%e 19: .1111.. 60
%e 20: .111111 63
%e 21: 1...... 64
%e 22: 1....11 67
%e (End)
%t Take[Nest[Union[Flatten[# /. {{i_Integer -> i}, {i_Integer -> 2 i + 1}, {i_Integer -> 4 i}}]] &, {1}, 5], 32] (* Or *)
%t Select[Range[124], FreeQ[Length /@ Select[Split[IntegerDigits[#, 2]], First[#] == 0 &], _?OddQ] &] (* _Birkas Gyorgy_, May 29 2012 *)
%o (Haskell)
%o import Data.Set (singleton, deleteFindMin, insert)
%o a060142 n = a060142_list !! n
%o a060142_list = 0 : f (singleton 1) where
%o f s = x : f (insert (4 * x) $ insert (2 * x + 1) s') where
%o (x, s') = deleteFindMin s
%o -- _Reinhard Zumkeller_, Nov 26 2012
%o (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 Cf. A219608 (odd terms), A060138, A060139, A060140, A060141, A086747.
%Y Cf. A003714 (no consecutive 1's in binary expansion).
%Y Odd partitions are counted by A000009.
%Y Numbers with an odd number of 1's in binary expansion are A000069.
%Y Numbers whose binary expansion has odd length are A053738.
%Y All of the following pertain to compositions in standard order (A066099):
%Y - Length is A000120.
%Y - Compositions without odd parts are A062880.
%Y - Sum is A070939.
%Y - Product is A124758.
%Y - Strict compositions are A233564.
%Y - Heinz number is A333219.
%Y - Number of distinct parts is A334028.
%K nonn
%O 0,3
%A _Clark Kimberling_, Mar 05 2001
%E Corrected by _T. D. Noe_, Nov 01 2006
%E Definition simplified by _Charles R Greathouse IV_, Oct 21 2013
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