A139351 Let the binary expansion of n be n = Sum_{k} 2^{r_k}, let e(n) be the number of r_k's that are even, o(n) the number that are odd; sequence gives e(n).

0, 1, 0, 1, 1, 2, 1, 2, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 3, 2, 3, 1, 2, 1, 2, 2, 3, 2, 3, 0, 1, 0, 1, 1, 2, 1, 2, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 3, 2, 3, 1, 2, 1, 2, 2, 3, 2, 3, 1, 2, 1, 2, 2, 3, 2, 3, 1, 2, 1, 2, 2, 3, 2, 3, 2, 3, 2, 3, 3, 4, 3, 4, 2, 3, 2, 3, 3, 4, 3, 4, 1, 2, 1

Offset

0,6

Comments

e(n)+o(n) = A000120(n), the binary weight of n.

a(n) is also number of 1's and 3's in 4-ary representation of n. - Frank Ruskey, May 02 2009

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Franklin T. Adams-Watters and Frank Ruskey, Generating Functions for the Digital Sum and Other Digit Counting Sequences, JIS 12 (2009), Article 09.5.6.

N. J. A. Sloane, Fortran program for this and related sequences.

Formula

a(n) + A139352(n) = A000120(n).

G.f.: (1/(1-z))*Sum_{m>=0} (z^(4^m)/(1+z^(4^m))). - Frank Ruskey, May 03 2009

Recurrence relation: a(0)=0, a(4m) = a(4m+2) = a(m), a(4m+1) = a(4m+3) = 1+a(m). - Frank Ruskey, May 11 2009

a(n) = Sum_{k} A030308(n,k)*A059841(k). - Philippe Deléham, Oct 14 2011

Example

For n = 43 = 2^0 + 2^1 + 2^3 + 2^5, e(43)=1, o(43)=3.

Maple

A139351 := proc(n)
    local a, bdgs, r;
    a := 0 ;
    bdgs := convert(n, base, 2) ;
    for r from 1 to nops(bdgs) by 2 do
        if op(r, bdgs) = 1 then
            a := a+1 ;
        end if;
    end do:
    a;
end proc: # R. J. Mathar, Jul 21 2016

Mathematica

terms = 99; s = (1/(1-z))*Sum[z^(4^m)/(1+z^(4^m)), {m, 0, Log[4, terms] // Ceiling}] + O[z]^terms; CoefficientList[s, z] (* Jean-François Alcover, Jul 21 2017 *)
a[0] = 0; a[n_] := a[n] = a[Floor[n/4]] + If[OddQ[Mod[n, 4]], 1, 0]; Array[a, 100, 0] (* Amiram Eldar, Jul 18 2023 *)

Prog

(Fortran) See Sloane link.
(Haskell)
import Data.List (unfoldr)
a139351 = sum . map (`mod` 2) .
   unfoldr (\x -> if x == 0 then Nothing else Just (x, x `div` 4)
-- Reinhard Zumkeller, Apr 22 2011
(PARI) a(n)=if(n>3, a(n\4))+n%2 \\ Charles R Greathouse IV, Apr 21 2016

Crossrefs

Cf. A000120, A030308, A059841, A139352, A139353, A139354, A139355, A039004, A139370, A139371, A139372, A139373.

Sequence in context: A345021 A104886 A215604 * A285677 A036578 A229764

Adjacent sequences: A139348 A139349 A139350 * A139352 A139353 A139354

Keyword

nonn,base,easy

Author

Nadia Heninger and N. J. A. Sloane, Jun 07 2008

Extensions

Typo in example fixed by Reinhard Zumkeller, Apr 22 2011

Status

approved