cp's OEIS Frontend

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

%I A059010 #70 Mar 23 2024 20:03:08
%S A059010 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 A059010 48,51,53,54,57,58,60,63,64,67,69,70,73,74,76,79,81,82,84,87,88,91,93,
%U A059010 94,97,98,100,103,104,107,109,110,112,115,117,118,121,122,124,127,129,130
%N A059010 Natural numbers having an even number of nonleading zeros in their binary expansion.
%C A059010 Positions of ones in A298952, and of zeros in A059448. - _John Keith_, Mar 09 2022
%H A059010 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 A059010 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 A059010 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 A059010 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 A059010 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 A059010 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A059010 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 A059010 Select[Range[130], EvenQ @ DigitCount[#, 2, 0] &] (* _Jean-François Alcover_, Apr 11 2011 *)
%o A059010 (PARI) is(n)=hammingweight(bitneg(n,#binary(n)))%2==0 \\ _Charles R Greathouse IV_, Mar 26 2013
%o A059010 (PARI) a(n) = if(n==0,1, 2*n + (logint(n,2) - hammingweight(n)) % 2); \\ _Kevin Ryde_, Mar 11 2021
%o A059010 (Haskell)
%o A059010 a059010 n = a059010_list !! (n-1)
%o A059010 a059010_list = filter (even . a023416) [1..]
%o A059010 -- _Reinhard Zumkeller_, Jan 21 2014
%o A059010 (Python)
%o A059010 #Program to generate the b-file
%o A059010 i=1
%o A059010 j=0
%o A059010 while j<=250:
%o A059010     if bin(i)[2:].count("0")%2==0:
%o A059010         print(str(j)+" "+str(i))
%o A059010         j+=1
%o A059010     i+=1 # _Indranil Ghosh_, Feb 03 2017
%o A059010 (R)
%o A059010 maxrow <- 4 # by choice
%o A059010 onezeros <- 1
%o A059010 for(m in 1:(maxrow+1)){
%o A059010   row <- onezeros[2^(m-1):(2^m-1)]
%o A059010   onezeros <- c(onezeros, c(1-row, row) )
%o A059010 }
%o A059010 a <- which(onezeros == 1)
%o A059010 a
%o A059010 # _Yosu Yurramendi_, Mar 28 2017
%Y A059010 Cf. A059009 (complement).
%Y A059010 Cf. A023416 (number of 0-bits), A059448 (their parity), A298952 (opposite parity).
%Y A059010 Cf.  A001969 (even 1-bits), A059012 (even both 0's and 1's), A059014 (even 0's, odd 1's).
%K A059010 nonn,easy,base,nice
%O A059010 0,2
%A A059010 _Patrick De Geest_, Dec 15 2000
%E A059010 Name clarified by _Antti Karttunen_, Mar 28 2017