A059010 Natural numbers having an even number of nonleading zeros in their binary expansion.

1, 3, 4, 7, 9, 10, 12, 15, 16, 19, 21, 22, 25, 26, 28, 31, 33, 34, 36, 39, 40, 43, 45, 46, 48, 51, 53, 54, 57, 58, 60, 63, 64, 67, 69, 70, 73, 74, 76, 79, 81, 82, 84, 87, 88, 91, 93, 94, 97, 98, 100, 103, 104, 107, 109, 110, 112, 115, 117, 118, 121, 122, 124, 127, 129, 130

Offset: 0

Author

Patrick De Geest, Dec 15 2000

Keywords

• nonn
• easy
• base
• nice

Comments

Positions of ones in A298952, and of zeros in A059448. - John Keith, Mar 09 2022

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..25000 (terms 0..1000 from T. D. Noe)
• Jean Paul Allouche, Jeffrey Shallit, and Guentcho Skordev, Self-generating sets, integers with missing blocks and substitutions, Discrete Math. 292 (2005) 1-15.
• Clark Kimberling, Affinely recursive sets and orderings of languages, Discrete Math., 274 (2004), 147-160. [From _N. J. A. Sloane_, Jan 31 2012]
• Jeffrey Shallit, Additive Number Theory via Automata and Logic, arXiv:2112.13627 [math.NT], 2021.
• Wadim Zudilin, A strange identity of an MF (Mahler function), arXiv:2403.13604 [math.NT], 2024.
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A059009 (complement).

Cf. A023416 (number of 0-bits), A059448 (their parity), A298952 (opposite parity).

Cf. A001969 (even 1-bits), A059012 (even both 0's and 1's), A059014 (even 0's, odd 1's).

Programs

Haskell

a059010 n = a059010_list !! (n-1)
a059010_list = filter (even . a023416) [1..]
-- Reinhard Zumkeller, Jan 21 2014

Mathematica

Select[Range[130], EvenQ @ DigitCount[#, 2, 0] &] (* Jean-François Alcover, Apr 11 2011 *)

PARI

is(n)=hammingweight(bitneg(n,#binary(n)))%2==0 \\ Charles R Greathouse IV, Mar 26 2013

PARI

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

Python

#Program to generate the b-file
i=1
j=0
while j<=250:
    if bin(i)[2:].count("0")%2==0:
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Feb 03 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 == 1)
a
# Yosu Yurramendi, Mar 28 2017

Formula

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

Extensions

Name clarified by Antti Karttunen, Mar 28 2017