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

Author

Clark Kimberling

Keywords

• nonn
• base

Examples

			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
		

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

Crossrefs

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

Programs

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

PARI

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

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

Extensions

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