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.
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
KEYWORD
nonn,base,easy
AUTHOR
STATUS
approved