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

0, 5, 9, 10, 11, 13, 17, 18, 19, 20, 22, 23, 25, 26, 27, 29, 33, 34, 35, 36, 38, 39, 40, 44, 46, 47, 49, 50, 51, 52, 54, 55, 57, 58, 59, 61, 65, 66, 67, 68, 70, 71, 72, 76, 78, 79, 80, 85, 88, 92, 94, 95, 97, 98, 99, 100, 102, 103, 104, 108, 110, 111, 113, 114

Offset: 0

Author

Vladimir Shevelev, Feb 04 2016

Keywords

• nonn
• base

Comments

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.

Examples

			In binary representation 19=10011 has an even number (two) of runs of 1's. So, 19 is a member.
		

Links

• Peter J. C. Moses (terms 0..999) & Antti Karttunen, Table of n, a(n) for n = 0..8256
• Vladimir Shevelev, Two analogs of Thue-Morse sequence, arXiv:1603.04434 [math.NT], 2016.

Crossrefs

Positions of even terms in A069010.

Cf. A010060, A001969, A268411.

Cf. A268415 (complement).

Cf. A268383 (the least monotonic left inverse).

Cf. A268476 (primes in this sequence).

Programs

Mathematica

balancedBinary:=Join[#,{0}]-Join[{0},#]&[IntegerDigits[#,2]]&;
Flatten[Position[Map[Mod[Count[balancedBinary[#],1],2]&,Range[0,100]],0,1]-1] (* Peter J. C. Moses, Feb 04 2016 *)

Python

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

Formula

Other identities. For all n >= 0:

A268383(a(n)) = n.

Extensions

More terms from Peter J. C. Moses, Feb 04 2016