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

Author

Zak Seidov, Aug 08 2006

Keywords

• nonn
• easy

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

Examples

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

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.

Crossrefs

Cf. A121538, A121540, A121541, A121542.

Programs

Haskell

import Data.List (elemIndices)
a121539 n = a121539_list !! (n-1)
a121539_list = elemIndices 1 a035263_list
-- Reinhard Zumkeller, Mar 01 2012

Magma

[n: n in [0..200] | Valuation(n+1, 2) mod 2 eq 0 ]; // Vincenzo Librandi, Apr 16 2015

Mathematica

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

PARI

is(n)=valuation(n+1,2)%2==0 \\ Charles R Greathouse IV, Sep 23 2012

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

Python

def A121539(n):
    def f(x):
        c, s = n+x, bin(x)[2:]
        l = len(s)
        for i in range(l&1^1,l,2):
            c -= int(s[i])+int('0'+s[:i],2)
        return c
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m-1 # Chai Wah Wu, Jan 29 2025

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

Extensions

Edited by N. J. A. Sloane at the suggestion of Stefan Steinerberger, Dec 17 2007