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%I #42 Jul 18 2023 02:32:11
%S 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,
%T 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,
%U 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
%N 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).
%C e(n)+o(n) = A000120(n), the binary weight of n.
%C a(n) is also number of 1's and 3's in 4-ary representation of n. - _Frank Ruskey_, May 02 2009
%H Reinhard Zumkeller, <a href="/A139351/b139351.txt">Table of n, a(n) for n = 0..10000</a>
%H Franklin T. Adams-Watters and Frank Ruskey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Ruskey2/ruskey14.html">Generating Functions for the Digital Sum and Other Digit Counting Sequences</a>, JIS 12 (2009), Article 09.5.6.
%H N. J. A. Sloane, <a href="/A139351/a139351.f.txt">Fortran program for this and related sequences</a>.
%F a(n) + A139352(n) = A000120(n).
%F G.f.: (1/(1-z))*Sum_{m>=0} (z^(4^m)/(1+z^(4^m))). - _Frank Ruskey_, May 03 2009
%F 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
%F a(n) = Sum_{k} A030308(n,k)*A059841(k). - _Philippe Deléham_, Oct 14 2011
%e For n = 43 = 2^0 + 2^1 + 2^3 + 2^5, e(43)=1, o(43)=3.
%p A139351 := proc(n)
%p local a,bdgs,r;
%p a := 0 ;
%p bdgs := convert(n,base,2) ;
%p for r from 1 to nops(bdgs) by 2 do
%p if op(r,bdgs) = 1 then
%p a := a+1 ;
%p end if;
%p end do:
%p a;
%p end proc: # _R. J. Mathar_, Jul 21 2016
%t 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 *)
%t 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 *)
%o (Fortran) See Sloane link.
%o (Haskell)
%o import Data.List (unfoldr)
%o a139351 = sum . map (`mod` 2) .
%o unfoldr (\x -> if x == 0 then Nothing else Just (x, x `div` 4)
%o -- _Reinhard Zumkeller_, Apr 22 2011
%o (PARI) a(n)=if(n>3,a(n\4))+n%2 \\ _Charles R Greathouse IV_, Apr 21 2016
%Y Cf. A000120, A030308, A059841, A139352, A139353, A139354, A139355, A039004, A139370, A139371, A139372, A139373.
%K nonn,base,easy
%O 0,6
%A _Nadia Heninger_ and _N. J. A. Sloane_, Jun 07 2008
%E Typo in example fixed by _Reinhard Zumkeller_, Apr 22 2011
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