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

0,2

Comments

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

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.

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

Example

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

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 *)

Prog

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

Crossrefs

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

Cf. A023416.

Sequence in context: A047267 A058591 A247062 * A214642 A026179 A300063

Adjacent sequences: A059006 A059007 A059008 * A059010 A059011 A059012

Keyword

nonn,base,easy,nice

Author

Patrick De Geest, Dec 15 2000

Status

approved