A059009 Numbers having an odd number of zeros in their binary expansion.

0, 2, 5, 6, 8, 11, 13, 14, 17, 18, 20, 23, 24, 27, 29, 30, 32, 35, 37, 38, 41, 42, 44, 47, 49, 50, 52, 55, 56, 59, 61, 62, 65, 66, 68, 71, 72, 75, 77, 78, 80, 83, 85, 86, 89, 90, 92, 95, 96, 99, 101, 102, 105, 106, 108, 111, 113, 114, 116, 119, 120, 123, 125, 126, 128, 131

Offset: 0

Author

Patrick De Geest, Dec 15 2000

Keywords

• nonn
• base
• easy
• nice

Comments

Positions of ones in A059448 for n >= 1. - John Keith, Mar 09 2022

Examples

			18 is in the sequence because 18 = 10010_2. '10010' has three zeros. - _Indranil Ghosh_, Feb 04 2017
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..25000 (terms 0..1000 from T. D. Noe)
• Jeffrey Shallit, Additive Number Theory via Automata and Logic, arXiv:2112.13627 [math.NT], 2021.

Crossrefs

Cf. A000069, A001969, A059010, A059011, A059012, A059013, A059014, A059448.

Cf. A023416.

Programs

Haskell

a059009 n = a059009_list !! (n-1)
a059009_list = filter (odd . a023416) [1..]
-- Reinhard Zumkeller, Jan 21 2014

Maple

a:= proc(n) option remember;
  if n::even then -a(n/2) + 3*n + 1 else a((n-1)/2) + n + 1 fi
end proc:
a(0):= 0:
seq(a(n),n=0..100); # Robert Israel, Feb 23 2016

Mathematica

Select[Range[0,150],OddQ[Count[IntegerDigits[#,2],0]]&] (* Harvey P. Dale, Oct 22 2011 *)

PARI

is(n)=hammingweight(bitneg(n,#binary(n)))%2 \\ Charles R Greathouse IV, Mar 26 2013

PARI

a(n) = if(n==0,0, 2*n + (logint(n,2) - hammingweight(n) + 1) % 2); \\ Kevin Ryde, Mar 11 2021

Python

i=j=0
while j<=800:
    if bin(i)[2:].count("0")%2:
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Feb 04 2017

R

maxrow <- 4 # by choice
onezeros <- 1
for(m in 1:(maxrow+1)){
  row <- onezeros[2^(m-1):(2^m-1)]
  onezeros <- c(onezeros, c(1-row, row) )
}
a <- which(onezeros == 0)
a
# Yosu Yurramendi, Mar 28 2017

Formula

a(0) = 0, a(2*n) = -a(n) + 6*n + 1, a(2*n+1) = a(n) + 2*n + 2. a(n) = 2*n + 1/2(1-(-1)^A023416(n)) = 2*n + A059448(n). - Ralf Stephan, Sep 17 2003