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A053754 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).

%I A053754 #45 Sep 20 2021 22:19:20
%S A053754 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 A053754 45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,128,129,130,
%U A053754 131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148
%N A053754 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 A053754 Runs of successive terms with same number of bits have length twice powers of 4 (A081294). [Clarified by _Michel Marcus_, Oct 21 2020]
%C A053754 The sequence A081294 counts compositions of even numbers - _Gus Wiseman_, Aug 12 2021
%C A053754 A031443 is a subsequence; A179888 is the intersection of this sequence and A032925. - _Reinhard Zumkeller_, Jul 31 2010
%C A053754 The lower and upper asymptotic densities of this sequence are 1/3 and 2/3, respectively. - _Amiram Eldar_, Feb 01 2021
%C A053754 From _Gus Wiseman_, Aug 10 2021: (Start)
%C A053754 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 A053754   0: ()   2: (2)      8: (4)
%C A053754           3: (1,1)    9: (3,1)
%C A053754                      10: (2,2)
%C A053754                      11: (2,1,1)
%C A053754                      12: (1,3)
%C A053754                      13: (1,2,1)
%C A053754                      14: (1,1,2)
%C A053754                      15: (1,1,1,1)
%C A053754 The following pertain to compositions in standard order: A000120, A029837, A070939, A066099, A124767.
%C A053754 (End)
%H A053754 Reinhard Zumkeller, <a href="/A053754/b053754.txt">Table of n, a(n) for n = 1..10001</a>
%t A053754 Select[Range[0, 150], EvenQ @ IntegerLength[#, 2] &] (* _Amiram Eldar_, Feb 01 2021 *)
%o A053754 (Haskell)
%o A053754 a053754 n = a053754_list !! (n-1)
%o A053754 a053754_list = 0 : filter (even . a070939) [1..]
%o A053754 -- _Reinhard Zumkeller_, Apr 18 2015
%o A053754 (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 A053754 (PARI) a(n) = n-1 + (1<<bitand(logint(6*n-3,2),-2))\3; \\ _Kevin Ryde_, Apr 30 2021
%Y A053754 Cf. A031443, A032925, A179888.
%Y A053754 Positions of even terms in A029837 with offset 0.
%Y A053754 The complement (the odd version) is A053738, counted by A000302.
%Y A053754 The version for Heinz numbers of partitions is A300061, counted by A058696.
%Y A053754 Cf. A000120, A001969, A008549, A070939, A080791, A114121.
%K A053754 base,easy,nonn
%O A053754 1,2
%A A053754 _Henry Bottomley_, Apr 06 2000
%E A053754 Offset corrected by _Reinhard Zumkeller_, Jul 30 2010