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

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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

Cf. A121538, A121540, A121541, A121542.

Sequence in context: A097274 A322572 A254438 * A257457 A122138 A047418

Adjacent sequences: A121536 A121537 A121538 * A121540 A121541 A121542

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