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If k is in the sequence then 2*k and 2*k+1 are not (and 0 is in the sequence); when written in binary k has an even number of bits (0 has 0 digits).
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%I #45 Sep 20 2021 22:19:20

%S 0,2,3,8,9,10,11,12,13,14,15,32,33,34,35,36,37,38,39,40,41,42,43,44,

%T 45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,128,129,130,

%U 131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148

%N If k is in the sequence then 2*k and 2*k+1 are not (and 0 is in the sequence); when written in binary k has an even number of bits (0 has 0 digits).

%C Runs of successive terms with same number of bits have length twice powers of 4 (A081294). [Clarified by _Michel Marcus_, Oct 21 2020]

%C The sequence A081294 counts compositions of even numbers - _Gus Wiseman_, Aug 12 2021

%C A031443 is a subsequence; A179888 is the intersection of this sequence and A032925. - _Reinhard Zumkeller_, Jul 31 2010

%C The lower and upper asymptotic densities of this sequence are 1/3 and 2/3, respectively. - _Amiram Eldar_, Feb 01 2021

%C From _Gus Wiseman_, Aug 10 2021: (Start)

%C Also numbers k such that the k-th composition in standard order (row k of A066099) has even sum. The terms and corresponding compositions begin:

%C 0: () 2: (2) 8: (4)

%C 3: (1,1) 9: (3,1)

%C 10: (2,2)

%C 11: (2,1,1)

%C 12: (1,3)

%C 13: (1,2,1)

%C 14: (1,1,2)

%C 15: (1,1,1,1)

%C The following pertain to compositions in standard order: A000120, A029837, A070939, A066099, A124767.

%C (End)

%H Reinhard Zumkeller, <a href="/A053754/b053754.txt">Table of n, a(n) for n = 1..10001</a>

%t Select[Range[0, 150], EvenQ @ IntegerLength[#, 2] &] (* _Amiram Eldar_, Feb 01 2021 *)

%o (Haskell)

%o a053754 n = a053754_list !! (n-1)

%o a053754_list = 0 : filter (even . a070939) [1..]

%o -- _Reinhard Zumkeller_, Apr 18 2015

%o (PARI) lista(nn) = {my(va = vector(nn)); for (n=2, nn, my(k=va[n-1]+1); while (#select(x->(x==k\2), va), k++); va[n] = k;); va;} \\ _Michel Marcus_, Oct 20 2020

%o (PARI) a(n) = n-1 + (1<<bitand(logint(6*n-3,2),-2))\3; \\ _Kevin Ryde_, Apr 30 2021

%Y Cf. A031443, A032925, A179888.

%Y Positions of even terms in A029837 with offset 0.

%Y The complement (the odd version) is A053738, counted by A000302.

%Y The version for Heinz numbers of partitions is A300061, counted by A058696.

%Y Cf. A000120, A001969, A008549, A070939, A080791, A114121.

%K base,easy,nonn

%O 1,2

%A _Henry Bottomley_, Apr 06 2000

%E Offset corrected by _Reinhard Zumkeller_, Jul 30 2010