A033265 Number of i such that d(i) >= d(i-1), where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.

0, 1, 1, 2, 1, 2, 2, 3, 2, 2, 2, 3, 2, 3, 3, 4, 3, 3, 3, 3, 2, 3, 3, 4, 3, 3, 3, 4, 3, 4, 4, 5, 4, 4, 4, 4, 3, 4, 4, 4, 3, 3, 3, 4, 3, 4, 4, 5, 4, 4, 4, 4, 3, 4, 4, 5, 4, 4, 4, 5, 4, 5, 5, 6, 5, 5, 5, 5, 4, 5, 5, 5, 4, 4, 4, 5, 4, 5, 5, 5, 4, 4, 4, 4, 3, 4, 4, 5, 4, 4

Offset

1,4

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Ralf Stephan, Some divide-and-conquer sequences ...

Ralf Stephan, Table of generating functions

Index entries for sequences related to binary expansion of n

Formula

From Ralf Stephan, Oct 05 2003: (Start)

a(0) = 0, a(2n) = a(n) + 1, a(2n+1) = a(n) + [n odd].

a(n) = A014081(n) + A023416(n).

G.f.: 1/(1-x) * Sum_{k>=0} (t^2 + t^3 + t^4)/((1+t)*(1+t^2)), t=x^2^k. (End)

a(n) = -1 + A297113(A005940(1+n)). - Antti Karttunen, Dec 30 2017

Example

The base-2 representation of n=4 is 100 with d(0)=0, d(1)=0, d(2)=1. There are two rise-or-equal, one from d(0) to d(1) and one from d(1) to d(2), so a(4)=2. - R. J. Mathar, Oct 16 2015

Maple

A033265 := proc(n)
    a := 0 ;
    dgs := convert(n, base, 2);
    for i from 2 to nops(dgs) do
        if op(i, dgs)>=op(i-1, dgs) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Oct 16 2015

Prog

(PARI) A033265(n) = { my(i=0); while(n>1, if((n%4)!=1, i++); n >>= 1); (i); }; \\ Antti Karttunen, Aug 06 2023

Crossrefs

Cf. A014081, A023416, A037800, A037809, A005940, A156552, A297113, A364567 [= 2^a(n)].

Sequence in context: A358371 A263922 A057526 * A096004 A193495 A071068

Adjacent sequences: A033262 A033263 A033264 * A033266 A033267 A033268

Keyword

nonn,base

Author

Clark Kimberling

Extensions

Sign in Name corrected by R. J. Mathar, Oct 16 2015

Status

approved