%I A023416
%S A023416 1,0,1,0,2,1,1,0,3,2,2,1,2,1,1,0,4,3,3,2,3,2,2,1,3,2,2,1,2,1,1,0,5,4,4,
%T A023416 3,4,3,3,2,4,3,3,2,3,2,2,1,4,3,3,2,3,2,2,1,3,2,2,1,2,1,1,0,6,5,5,4,5,4,
%U A023416 4,3,5,4,4,3,4,3,3,2,5,4,4,3,4,3,3,2,4,3,3,2,3,2,2,1,5,4,4,3,4,3,3,2,4
%N A023416 Number of 0's in binary expansion of n.
%C A023416 Another version (A080791) has a(0) = 0.
%H A023416 N. J. A. Sloane, <a href="b023416.txt">Table of n, a(n) for n = 0..10000</
               a>
%H A023416 R. Stephan, <a href="somedcgf.html">Some divide-and-conquer sequences 
               ...</a>
%H A023416 R. Stephan, <a href="a079944.ps">Table of generating functions</a>
%H A023416 R. Stephan, <a href="http://arXiv.org/abs/math.CO/0307027">Divide-and-conquer 
               generating functions. I. Elementary sequences</a>
%H A023416 <a href="Sindx_Bi.html#binary">Index entries for sequences related to 
               binary expansion of n</a>
%F A023416 a(n) = 1, if n = 0; 0, if n = 1; a(n/2)+1 if n even; a((n-1)/2) if n 
               odd.
%F A023416 a(n) = 1 - (n mod 2) + a(floor(n/2)) - Marc LeBrun (mlb(AT)well.com), 
               Jul 12 2001
%F A023416 G.f.: 1 + 1/(1-x) * Sum(k>=0, x^(2^(k+1))/(1+x^2^k)). - Ralf Stephan 
               (ralf(AT)ark.in-berlin.de), Apr 15 2002
%t A023416 Table[ Count[ IntegerDigits[n, 2], 0], {n, 0, 100} ]
%Y A023416 The basic sequences concerning the binary expansion of n are A000120, 
               A000788, A000069, A001969, A023416, A059015.
%Y A023416 a(n) = A070939(n)-A000120(n). Also A008687(n+1) - 1.
%Y A023416 With initial zero and shifted right, same as A080791.
%Y A023416 a(n) = A000120(A035327(n)).
%Y A023416 Sequence in context: A126258 A116382 A050606 this_sequence A080791 A124748 
               A161225
%Y A023416 Adjacent sequences: A023413 A023414 A023415 this_sequence A023417 A023418 
               A023419
%K A023416 nonn,nice,easy,base
%O A023416 0,5
%A A023416 David W. Wilson (davidwwilson(AT)comcast.net)