A059010 - OEIS

%I #70 Mar 23 2024 20:03:08

%S 1,3,4,7,9,10,12,15,16,19,21,22,25,26,28,31,33,34,36,39,40,43,45,46,

%T 48,51,53,54,57,58,60,63,64,67,69,70,73,74,76,79,81,82,84,87,88,91,93,

%U 94,97,98,100,103,104,107,109,110,112,115,117,118,121,122,124,127,129,130

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

%C Positions of ones in A298952, and of zeros in A059448. - _John Keith_, Mar 09 2022

%H Indranil Ghosh, <a href="/A059010/b059010.txt">Table of n, a(n) for n = 0..25000</a> (terms 0..1000 from T. D. Noe)

%H Jean Paul Allouche, Jeffrey Shallit, and Guentcho Skordev, <a href="http://dx.doi.org/10.1016/j.disc.2004.12.004">Self-generating sets, integers with missing blocks and substitutions</a>, Discrete Math. 292 (2005) 1-15.

%H Clark Kimberling, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00085-2">Affinely recursive sets and orderings of languages</a>, Discrete Math., 274 (2004), 147-160. [From _N. J. A. Sloane_, Jan 31 2012]

%H Jeffrey Shallit, <a href="https://arxiv.org/abs/2112.13627">Additive Number Theory via Automata and Logic</a>, arXiv:2112.13627 [math.NT], 2021.

%H Wadim Zudilin, <a href="https://arxiv.org/abs/2403.13604">A strange identity of an MF (Mahler function)</a>, arXiv:2403.13604 [math.NT], 2024.

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%F 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

%t Select[Range[130], EvenQ @ DigitCount[#, 2, 0] &] (* _Jean-François Alcover_, Apr 11 2011 *)

%o (PARI) is(n)=hammingweight(bitneg(n,#binary(n)))%2==0 \\ _Charles R Greathouse IV_, Mar 26 2013

%o (PARI) a(n) = if(n==0,1, 2*n + (logint(n,2) - hammingweight(n)) % 2); \\ _Kevin Ryde_, Mar 11 2021

%o (Haskell)

%o a059010 n = a059010_list !! (n-1)

%o a059010_list = filter (even . a023416) [1..]

%o -- _Reinhard Zumkeller_, Jan 21 2014

%o (Python)

%o #Program to generate the b-file

%o i=1

%o j=0

%o while j<=250:

%o     if bin(i)[2:].count("0")%2==0:

%o         print(str(j)+" "+str(i))

%o         j+=1

%o     i+=1 # _Indranil Ghosh_, Feb 03 2017

%o (R)

%o maxrow <- 4 # by choice

%o onezeros <- 1

%o for(m in 1:(maxrow+1)){

%o   row <- onezeros[2^(m-1):(2^m-1)]

%o   onezeros <- c(onezeros, c(1-row, row) )

%o }

%o a <- which(onezeros == 1)

%o a

%o # _Yosu Yurramendi_, Mar 28 2017

%Y Cf. A059009 (complement).

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

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

%K nonn,easy,base,nice

%O 0,2

%A _Patrick De Geest_, Dec 15 2000

%E Name clarified by _Antti Karttunen_, Mar 28 2017