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A268412 Balanced evil numbers: numbers with an even number of runs of 1's in their binary expansion.

%I A268412 #36 Feb 11 2017 02:26:22
%S A268412 0,5,9,10,11,13,17,18,19,20,22,23,25,26,27,29,33,34,35,36,38,39,40,44,
%T A268412 46,47,49,50,51,52,54,55,57,58,59,61,65,66,67,68,70,71,72,76,78,79,80,
%U A268412 85,88,92,94,95,97,98,99,100,102,103,104,108,110,111,113,114
%N A268412 Balanced evil numbers: numbers with an even number of runs of 1's in their binary expansion.
%C A268412 In balanced binary system the sequence A268411 plays role of Thue-Morse sequence (A010060). Therefore, we call the balanced evil numbers those numbers n for which A268411(n) = 0.
%H A268412 Peter J. C. Moses (terms 0..999) & Antti Karttunen, <a href="/A268412/b268412.txt">Table of n, a(n) for n = 0..8256</a>
%H A268412 Vladimir Shevelev, <a href="http://arxiv.org/abs/1603.04434">Two analogs of Thue-Morse sequence</a>, arXiv:1603.04434 [math.NT], 2016.
%F A268412 Other identities. For all n >= 0:
%F A268412 A268383(a(n)) = n.
%e A268412 In binary representation 19=10011 has an even number (two) of runs of 1's. So, 19 is a member.
%t A268412 balancedBinary:=Join[#,{0}]-Join[{0},#]&[IntegerDigits[#,2]]&;
%t A268412 Flatten[Position[Map[Mod[Count[balancedBinary[#],1],2]&,Range[0,100]],0,1]-1] (* _Peter J. C. Moses_, Feb 04 2016 *)
%o A268412 (Python)
%o A268412 A268412_list = [i for i in range(10**6) if not len(list(filter(bool,format(i,'b').split('0')))) % 2] # _Chai Wah Wu_, Mar 01 2016
%Y A268412 Positions of even terms in A069010.
%Y A268412 Cf. A010060, A001969, A268411.
%Y A268412 Cf. A268415 (complement).
%Y A268412 Cf. A268383 (the least monotonic left inverse).
%Y A268412 Cf. A268476 (primes in this sequence).
%K A268412 nonn,base
%O A268412 0,2
%A A268412 _Vladimir Shevelev_, Feb 04 2016
%E A268412 More terms from _Peter J. C. Moses_, Feb 04 2016