A056973 Number of blocks of {0,0} in the binary expansion of n.

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 3, 2, 1, 1, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 4, 3, 2, 2, 2, 1, 1, 1, 2, 1, 0, 0, 1, 0, 0, 0, 3, 2, 1, 1, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 5, 4, 3, 3, 3, 2, 2, 2, 3, 2, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 4, 3, 2, 2, 2, 1, 1

Offset

1,8

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Jean-Paul Allouche and Jeffrey Shallit, Sums of digits and the Hurwitz zeta function, in: K. Nagasaka and E. Fouvry (eds.), Analytic Number Theory, Lecture Notes in Mathematics, Vol. 1434, Springer, Berlin, Heidelberg, 1990, pp. 19-30.

Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.

Ralf Stephan, Table of generating functions.

Eric Weisstein's World of Mathematics, Digit Block.

Index entries for sequences related to binary expansion of n

Formula

a(2n) = a(n) + [n is even], a(2n+1) = a(n).

G.f.: 1/(1-x) * Sum_{k>=0} t^4/((1+t)*(1+t^2)) where t=x^(2^k). - Ralf Stephan, Sep 10 2003

a(n) = A023416(n) - A033264(n). - Ralf Stephan, Sep 10 2003

Sum_{n>=1} a(n)/(n*(n+1)) = 2 - 3*log(2)/2 - Pi/4 (Allouche and Shallit, 1990). - Amiram Eldar, Jun 01 2021

Maple

f:= proc(n) option remember;
     if n mod 4 = 0 then 1 + procname(n/2)
     else procname(floor(n/2))
     fi
end proc:
f(1):= 0:
map(f, [$1..200]); # Robert Israel, Sep 02 2015

Mathematica

f[n_] := Count[Partition[IntegerDigits[n, 2], 2, 1], {0, 0}]; Table[f@ n, {n, 0, 102}] (* Michael De Vlieger, Sep 01 2015, after Robert G. Wilson v at A014081 *)
SequenceCount[#, {0, 0}, Overlaps->True]&/@(IntegerDigits[#, 2]&/@Range[0, 120]) (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 24 2018 *)

Prog

(Haskell)
a056973 = f 0 where
   f y x = if x == 0 then y else f (y + 0 ^ (mod x 4)) $ div x 2
-- Reinhard Zumkeller, Mar 31 2015
(PARI)
a(n) = { my(x = bitor(n, n>>1));
         if (x == 0, 0, 1 + logint(x, 2) - hammingweight(x)) }
vector(102, i, a(i))  \\ Gheorghe Coserea, Sep 01 2015

Crossrefs

Cf. A003754, A014081, A023416, A033264, A037800.

Cf. A107782.

Sequence in context: A373968 A324192 A108730 * A107782 A368413 A086017

Adjacent sequences: A056970 A056971 A056972 * A056974 A056975 A056976

Keyword

nonn,base

Author

Eric W. Weisstein

Status

approved