A139352 - OEIS

%I #44 Dec 19 2024 11:41:30

%S 0,0,1,1,0,0,1,1,1,1,2,2,1,1,2,2,0,0,1,1,0,0,1,1,1,1,2,2,1,1,2,2,1,1,

%T 2,2,1,1,2,2,2,2,3,3,2,2,3,3,1,1,2,2,1,1,2,2,2,2,3,3,2,2,3,3,0,0,1,1,

%U 0,0,1,1,1,1,2,2,1,1,2,2,0,0,1,1,0,0,1,1,1,1,2,2,1,1,2,2,1,1,2

%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 o(n).

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

%C a(n) is also the number of 2's and 3's in the 4-ary representation of n. - _Frank Ruskey_, May 02 2009

%H Reinhard Zumkeller, <a href="/A139352/b139352.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.

%F G.f.: (1/(1-z))*Sum_{m>=0} (z^(2*4^m)/(1+(2*4^m))). - _Frank Ruskey_, May 03 2009

%F Recurrence relation: a(0)=0, a(4m) = a(4m+1) = a(m), a(4m+2) = a(4m+3) = 1+a(m). - _Frank Ruskey_, May 11 2009

%F a(n) = Sum_{k} A030308(n,k)*A000035(k). - _Philippe Deléham_, Oct 14 2011

%e For n = 43 = 2^0 + 2^1 + 2^3 + 2^5, e(43)=1, o(43)=3. [Typo fixed by _Reinhard Zumkeller_, Apr 22 2011]

%p A139352 := proc(n)

%p     local a,bdgs,r;

%p     a := 0 ;

%p     bdgs := convert(n,base,2) ;

%p     for r from 2 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 a[n_] := Count[Position[Reverse@IntegerDigits[n, 2], 1]-1, {_?OddQ}];

%t Table[a[n], {n, 0, 99}] (* _Jean-François Alcover_, Mar 04 2023 *)

%t a[0] = 0; a[n_] := a[n] = a[Floor[n/4]] + If[Mod[n, 4] > 1, 1, 0]; Array[a, 100, 0] (* _Amiram Eldar_, Jul 18 2023 *)

%o (Fortran) c See link in A139351

%o (Haskell)

%o import Data.List (unfoldr)

%o a139352 = sum . map ((`div` 2) . (`mod` 4)) .

%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%4\2 \\ _Charles R Greathouse IV_, Apr 21 2016

%Y Cf. A000035, A000120, A030308, A039004, A139351, A139353, A139354, A139355, A139370, A139371, A139372, A139373.

%K nonn,base,easy

%O 0,11

%A _Nadia Heninger_ and _N. J. A. Sloane_, Jun 07 2008