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