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A121539 Numbers whose binary expansion ends in an even number of 1's. 24
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, 38, 40, 42, 43, 44, 46, 47, 48, 50, 51, 52, 54, 56, 58, 59, 60, 62, 63, 64, 66, 67, 68, 70, 72, 74, 75, 76, 78, 79, 80, 82, 83, 84, 86, 88, 90, 91, 92, 94, 96, 98, 99, 100 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Equivalently, increasing sequence defined by: "if k appears a*k+b does not", case a(1)=0, a=2, b=1.
Every even number ends with zero 1's and zero is even, so every even number is a term.
Consists of all even numbers together with A131323.
A035263(a(n)) = 1. - Reinhard Zumkeller, Mar 01 2012
LINKS
Narad Rampersad, Manon Stipulanti, The Formal Inverse of the Period-Doubling Sequence, arXiv:1807.11899 [math.CO], 2018.
FORMULA
A010060(a(n)) + A010060(a(n)+1) = 1. - Vladimir Shevelev, Jun 16 2009
a(n) = A003159(n) - 1. - Reinhard Zumkeller, Mar 01 2012
a(n) = (3/2)*n + O(log n). - Charles R Greathouse IV, Sep 23 2012
EXAMPLE
11 in binary is 1011, which ends with two 1's.
MATHEMATICA
s={2}; With[{a=2, b=1}, Do[If[FreeQ[s, (n-b)/a], AppendTo[s, n]], {n, 3, 100}]]; s
PROG
(Haskell)
import Data.List (elemIndices)
a121539 n = a121539_list !! (n-1)
a121539_list = elemIndices 1 a035263_list
-- Reinhard Zumkeller, Mar 01 2012
(PARI) is(n)=valuation(n+1, 2)%2==0 \\ Charles R Greathouse IV, Sep 23 2012
(Magma) [n: n in [0..200] | Valuation(n+1, 2) mod 2 eq 0 ]; // Vincenzo Librandi, Apr 16 2015
(Python)
def ok(n): b = bin(n)[2:]; return (len(b) - len(b.rstrip('1')))%2 == 0
print(list(filter(ok, range(101)))) # Michael S. Branicky, Jun 18 2021
CROSSREFS
Sequence in context: A097274 A322572 A254438 * A257457 A122138 A047418
KEYWORD
nonn,easy
AUTHOR
Zak Seidov, Aug 08 2006
EXTENSIONS
Edited by N. J. A. Sloane at the suggestion of Stefan Steinerberger, Dec 17 2007
STATUS
approved

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Last modified May 29 11:04 EDT 2024. Contains 372938 sequences. (Running on oeis4.)