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

0,2

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

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

Mathematica

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

Prog

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

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).

Sequence in context: A263488 A344416 A330178 * A066928 A032726 A029739

Adjacent sequences: A059007 A059008 A059009 * A059011 A059012 A059013

Keyword

nonn,easy,base,nice

Author

Patrick De Geest, Dec 15 2000

Extensions

Name clarified by Antti Karttunen, Mar 28 2017

Status

approved