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Numbers whose binary expansion ends in an even number of 1's.
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%I #31 Sep 08 2022 08:45:27

%S 0,2,3,4,6,8,10,11,12,14,15,16,18,19,20,22,24,26,27,28,30,32,34,35,36,

%T 38,40,42,43,44,46,47,48,50,51,52,54,56,58,59,60,62,63,64,66,67,68,70,

%U 72,74,75,76,78,79,80,82,83,84,86,88,90,91,92,94,96,98,99,100

%N Numbers whose binary expansion ends in an even number of 1's.

%C Equivalently, increasing sequence defined by: "if k appears a*k+b does not", case a(1)=0, a=2, b=1.

%C Every even number ends with zero 1's and zero is even, so every even number is a term.

%C Consists of all even numbers together with A131323.

%C A035263(a(n)) = 1. - _Reinhard Zumkeller_, Mar 01 2012

%H Reinhard Zumkeller, <a href="/A121539/b121539.txt">Table of n, a(n) for n = 1..10000</a>

%H Narad Rampersad, Manon Stipulanti, <a href="https://arxiv.org/abs/1807.11899">The Formal Inverse of the Period-Doubling Sequence</a>, arXiv:1807.11899 [math.CO], 2018.

%F A010060(a(n)) + A010060(a(n)+1) = 1. - _Vladimir Shevelev_, Jun 16 2009

%F a(n) = A003159(n) - 1. - _Reinhard Zumkeller_, Mar 01 2012

%F a(n) = (3/2)*n + O(log n). - _Charles R Greathouse IV_, Sep 23 2012

%e 11 in binary is 1011, which ends with two 1's.

%t s={2}; With[{a=2,b=1},Do[If[FreeQ[s,(n-b)/a],AppendTo[s,n]],{n,3,100}]];s

%o (Haskell)

%o import Data.List (elemIndices)

%o a121539 n = a121539_list !! (n-1)

%o a121539_list = elemIndices 1 a035263_list

%o -- _Reinhard Zumkeller_, Mar 01 2012

%o (PARI) is(n)=valuation(n+1,2)%2==0 \\ _Charles R Greathouse IV_, Sep 23 2012

%o (Magma) [n: n in [0..200] | Valuation(n+1, 2) mod 2 eq 0 ]; // _Vincenzo Librandi_, Apr 16 2015

%o (Python)

%o def ok(n): b = bin(n)[2:]; return (len(b) - len(b.rstrip('1')))%2 == 0

%o print(list(filter(ok, range(101)))) # _Michael S. Branicky_, Jun 18 2021

%Y Cf. A121538, A121540, A121541, A121542.

%K nonn,easy

%O 1,2

%A _Zak Seidov_, Aug 08 2006

%E Edited by _N. J. A. Sloane_ at the suggestion of _Stefan Steinerberger_, Dec 17 2007