cp's OEIS Frontend

A121539 Numbers whose binary expansion ends in an even number of 1's.

%I A121539 #35 Jan 29 2025 14:25:58
%S A121539 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 A121539 38,40,42,43,44,46,47,48,50,51,52,54,56,58,59,60,62,63,64,66,67,68,70,
%U A121539 72,74,75,76,78,79,80,82,83,84,86,88,90,91,92,94,96,98,99,100
%N A121539 Numbers whose binary expansion ends in an even number of 1's.
%C A121539 Equivalently, increasing sequence defined by: "if k appears a*k+b does not", case a(1)=0, a=2, b=1.
%C A121539 Every even number ends with zero 1's and zero is even, so every even number is a term.
%C A121539 Consists of all even numbers together with A131323.
%C A121539 A035263(a(n)) = 1. - _Reinhard Zumkeller_, Mar 01 2012
%H A121539 Reinhard Zumkeller, <a href="/A121539/b121539.txt">Table of n, a(n) for n = 1..10000</a>
%H A121539 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 A121539 A010060(a(n)) + A010060(a(n)+1) = 1. - _Vladimir Shevelev_, Jun 16 2009
%F A121539 a(n) = A003159(n) - 1. - _Reinhard Zumkeller_, Mar 01 2012
%F A121539 a(n) = (3/2)*n + O(log n). - _Charles R Greathouse IV_, Sep 23 2012
%e A121539 11 in binary is 1011, which ends with two 1's.
%t A121539 s={2}; With[{a=2,b=1},Do[If[FreeQ[s,(n-b)/a],AppendTo[s,n]],{n,3,100}]];s
%o A121539 (Haskell)
%o A121539 import Data.List (elemIndices)
%o A121539 a121539 n = a121539_list !! (n-1)
%o A121539 a121539_list = elemIndices 1 a035263_list
%o A121539 -- _Reinhard Zumkeller_, Mar 01 2012
%o A121539 (PARI) is(n)=valuation(n+1,2)%2==0 \\ _Charles R Greathouse IV_, Sep 23 2012
%o A121539 (Magma) [n: n in [0..200] | Valuation(n+1, 2) mod 2 eq 0 ]; // _Vincenzo Librandi_, Apr 16 2015
%o A121539 (Python)
%o A121539 def ok(n): b = bin(n)[2:]; return (len(b) - len(b.rstrip('1')))%2 == 0
%o A121539 print(list(filter(ok, range(101)))) # _Michael S. Branicky_, Jun 18 2021
%o A121539 (Python)
%o A121539 def A121539(n):
%o A121539     def f(x):
%o A121539         c, s = n+x, bin(x)[2:]
%o A121539         l = len(s)
%o A121539         for i in range(l&1^1,l,2):
%o A121539             c -= int(s[i])+int('0'+s[:i],2)
%o A121539         return c
%o A121539     m, k = n, f(n)
%o A121539     while m != k: m, k = k, f(k)
%o A121539     return m-1 # _Chai Wah Wu_, Jan 29 2025
%Y A121539 Cf. A121538, A121540, A121541, A121542.
%K A121539 nonn,easy
%O A121539 1,2
%A A121539 _Zak Seidov_, Aug 08 2006
%E A121539 Edited by _N. J. A. Sloane_ at the suggestion of _Stefan Steinerberger_, Dec 17 2007