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A023416
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Number of 0's in binary expansion of n.
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206
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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, 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, 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
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OFFSET
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0,5
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COMMENTS
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Another version (A080791) has a(0) = 0.
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n = 0..10000
F. T. Adams-Watters, F. Ruskey, Generating Functions for the Digital Sum and Other Digit Counting Sequences, JIS 12 (2009) 09.5.6
J.-P. Allouche, J. O. Shallit, Infinite products associated with counting blocks in binary strings, J. London Math. Soc.39 (1989) 193-204.
K. Hessami Pilehrood, T. Hessami Pilehrood, Vacca-Type Series for Values of the Generalized Euler Constant Function and its Derivative, J. Integer Sequences, 13 (2010), #10.7.3.
Vladimir Shevelev, The number of permutations with prescribed up-down structure as a function of two variables, INTEGERS, 12 (2012), #A1. - From N. J. A. Sloane, Feb 07 2013
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
Index entries for sequences related to binary expansion of n
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FORMULA
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a(n) = 1, if n = 0; 0, if n = 1; a(n/2)+1 if n even; a((n-1)/2) if n odd.
a(n) = 1 - (n mod 2) + a(floor(n/2)). - Marc LeBrun, Jul 12 2001
G.f.: 1 + 1/(1-x) * Sum_{k>=0} x^(2^(k+1))/(1+x^2^k). - Ralf Stephan, Apr 15 2002
a(n) = A070939(n) - A000120(n).
a(n) = A008687(n+1) - 1.
a(n) = A000120(A035327(n)).
From Hieronymus Fischer, Jun 12 2012: (Start)
a(n) = m + 1 + Sum_{j=1..m+1} (floor(n/2^j) - floor(n/2^j + 1/2)), where m=floor(log_2(n)).
General formulas for the number of digits <= d in the base p representation n, where 0 <= d < p.
a(n) = m + 1 + Sum_{j=1..m+1} (floor(n/p^j) - floor(n/p^j + (p-d-1)/p)), where m=floor(log_p(n)).
G.f.: g(x) = 1 + (1/(1-x))*Sum_{j>=0} (1-x^(d*p^j))*x^p^j) + (1-x^p^j)*x^p^(j+1)/(1-x^p^(j+1)). (End)
Product_{n>=1} ((2*n)/(2*n+1))^((-1)^a(n)) = sqrt(2)/2 (A010503) (see Allouche & Shallit link). - Michel Marcus, Aug 31 2014
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MAPLE
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A023416 := proc(n)
if n = 0 then
1;
else
add(1-e, e=convert(n, base, 2)) ;
end if;
end proc: # R. J. Mathar, Jul 21 2012
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MATHEMATICA
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Table[ Count[ IntegerDigits[n, 2], 0], {n, 0, 100} ]
DigitCount[Range[0, 110], 2, 0] (* Harvey P. Dale, Jan 10 2013 *)
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PROG
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(Haskell)
a023416 0 = 1
a023416 1 = 0
a023416 n = a023416 n' + 1 - m where (n', m) = divMod n 2
a023416_list = 1 : c [0] where c (z:zs) = z : c (zs ++ [z+1, z])
-- Reinhard Zumkeller, Feb 19 2012, Jun 16 2011, Mar 07 2011
(PARI) a(n)=if(n==0, 1, n=binary(n); sum(i=1, #n, !n[i])) \\ Charles R Greathouse IV, Jun 10 2011
(PARI) a(n)=if(n==0, 1, #binary(n)-hammingweight(n)) \\ Charles R Greathouse IV, Nov 20 2012
(PARI) a(n) = if(n == 0, 1, 1+logint(n, 2) - hammingweight(n)) \\ Gheorghe Coserea, Sep 01 2015
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CROSSREFS
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The basic sequences concerning the binary expansion of n are A000120, A000788, A000069, A001969, A023416, A059015, A070939, A083652. Partial sums see A059015.
With initial zero and shifted right, same as A080791.
Cf. A055641 (for base 10).
Sequence in context: A116382 A050606 A277721 * A080791 A336361 A334204
Adjacent sequences: A023413 A023414 A023415 * A023417 A023418 A023419
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KEYWORD
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nonn,nice,easy,base
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AUTHOR
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David W. Wilson
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STATUS
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approved
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