A033264 Number of blocks of {1,0} in the binary expansion of n.

0, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 2, 2, 2, 2, 1, 2, 2, 3, 2, 3, 3, 3, 2, 2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 2, 1, 2, 2, 2

Offset

1,10

Comments

Number of i such that d(i) < d(i-1), where Sum_{d(i)*2^i: i=0,1,....,m} is base 2 representation of n.

This is the base-2 down-variation sequence; see A297330. - Clark Kimberling, Jan 18 2017

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

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

a(n) = A069010(n) - (n mod 2). - Ralf Stephan, Sep 10 2003

a(4n) = a(4n+1) = a(2n), a(4n+2) = a(n)+1, a(4n+3) = a(n). - Ralf Stephan, Aug 20 2003

a(n) = A087116(n) for n > 0, since strings of 0's alternate with strings of 1's, which end in (1,0). - Jonathan Sondow, Jan 17 2016

Sum_{n>=1} a(n)/(n*(n+1)) = Pi/4 - log(2)/2 (A196521) (Allouche and Shallit, 1990). - Amiram Eldar, Jun 01 2021

Maple

f:= proc(n) option remember; local k;
k:= n mod 4;
if k = 2 then procname((n-2)/4) + 1
elif k = 3 then procname((n-3)/4)
else procname((n-k)/2)
fi
end proc:
f(1):= 0: f(0):= q:
seq(f(i), i=1..100); # Robert Israel, Aug 31 2015

Mathematica

Table[Count[Partition[IntegerDigits[n, 2], 2, 1], {1, 0}], {n, 102}] (* Michael De Vlieger, Aug 31 2015, after Robert G. Wilson v at A014081 *)
Table[SequenceCount[IntegerDigits[n, 2], {1, 0}], {n, 110}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 26 2017 *)

Prog

(Haskell)
a033264 = f 0 . a030308_row where
   f c [] = c
   f c (0 : 1 : bs) = f (c + 1) bs
   f c (_ : bs) = f c bs
-- Reinhard Zumkeller, Feb 20 2014, Jun 17 2012
(PARI)
a(n) = { hammingweight(bitand(n>>1, bitneg(n))) }; \\ Gheorghe Coserea, Aug 30 2015

Crossrefs

Cf. A014081, A014082, A037800, A056974, A056975, A056976, A056977, A056978, A056979, A056980, A196521.

a(n) = A005811(n) - ceiling(A005811(n)/2) = A005811(n) - A069010(n).

Equals (A072219(n+1)-1)/2.

Cf. also A175047, A030308.

Essentially the same as A087116.

Sequence in context: A047988 A037818 A087116 * A258045 A239302 A256983

Adjacent sequences: A033261 A033262 A033263 * A033265 A033266 A033267

Keyword

nonn,base,easy

Author

Clark Kimberling

Status

approved