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
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.
Programs
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Haskell
import Data.List (elemIndices) a121539 n = a121539_list !! (n-1) a121539_list = elemIndices 1 a035263_list -- Reinhard Zumkeller, Mar 01 2012
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Magma
[n: n in [0..200] | Valuation(n+1, 2) mod 2 eq 0 ]; // Vincenzo Librandi, Apr 16 2015
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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
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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
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
Comments