cp's OEIS Frontend

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

%I A059009 #49 Mar 10 2022 14:18:14
%S A059009 0,2,5,6,8,11,13,14,17,18,20,23,24,27,29,30,32,35,37,38,41,42,44,47,
%T A059009 49,50,52,55,56,59,61,62,65,66,68,71,72,75,77,78,80,83,85,86,89,90,92,
%U A059009 95,96,99,101,102,105,106,108,111,113,114,116,119,120,123,125,126,128,131
%N A059009 Numbers having an odd number of zeros in their binary expansion.
%C A059009 Positions of ones in A059448 for n >= 1. - _John Keith_, Mar 09 2022
%H A059009 Indranil Ghosh, <a href="/A059009/b059009.txt">Table of n, a(n) for n = 0..25000</a> (terms 0..1000 from T. D. Noe)
%H A059009 Jeffrey Shallit, <a href="https://arxiv.org/abs/2112.13627">Additive Number Theory via Automata and Logic</a>, arXiv:2112.13627 [math.NT], 2021.
%F A059009 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
%e A059009 18 is in the sequence because 18 = 10010_2. '10010' has three zeros. - _Indranil Ghosh_, Feb 04 2017
%p A059009 a:= proc(n) option remember;
%p A059009   if n::even then -a(n/2) + 3*n + 1 else a((n-1)/2) + n + 1 fi
%p A059009 end proc:
%p A059009 a(0):= 0:
%p A059009 seq(a(n),n=0..100); # _Robert Israel_, Feb 23 2016
%t A059009 Select[Range[0,150],OddQ[Count[IntegerDigits[#,2],0]]&] (* _Harvey P. Dale_, Oct 22 2011 *)
%o A059009 (PARI) is(n)=hammingweight(bitneg(n,#binary(n)))%2 \\ _Charles R Greathouse IV_, Mar 26 2013
%o A059009 (PARI) a(n) = if(n==0,0, 2*n + (logint(n,2) - hammingweight(n) + 1) % 2); \\ _Kevin Ryde_, Mar 11 2021
%o A059009 (Haskell)
%o A059009 a059009 n = a059009_list !! (n-1)
%o A059009 a059009_list = filter (odd . a023416) [1..]
%o A059009 -- _Reinhard Zumkeller_, Jan 21 2014
%o A059009 (Python)
%o A059009 i=j=0
%o A059009 while j<=800:
%o A059009     if bin(i)[2:].count("0")%2:
%o A059009         print(str(j)+" "+str(i))
%o A059009         j+=1
%o A059009     i+=1 # _Indranil Ghosh_, Feb 04 2017
%o A059009 (R)
%o A059009 maxrow <- 4 # by choice
%o A059009 onezeros <- 1
%o A059009 for(m in 1:(maxrow+1)){
%o A059009   row <- onezeros[2^(m-1):(2^m-1)]
%o A059009   onezeros <- c(onezeros, c(1-row, row) )
%o A059009 }
%o A059009 a <- which(onezeros == 0)
%o A059009 a
%o A059009 # _Yosu Yurramendi_, Mar 28 2017
%Y A059009 Cf. A000069, A001969, A059010, A059011, A059012, A059013, A059014, A059448.
%Y A059009 Cf. A023416.
%K A059009 nonn,base,easy,nice
%O A059009 0,2
%A A059009 _Patrick De Geest_, Dec 15 2000