A063694 Remove odd-positioned bits from the binary expansion of n.

0, 1, 0, 1, 4, 5, 4, 5, 0, 1, 0, 1, 4, 5, 4, 5, 16, 17, 16, 17, 20, 21, 20, 21, 16, 17, 16, 17, 20, 21, 20, 21, 0, 1, 0, 1, 4, 5, 4, 5, 0, 1, 0, 1, 4, 5, 4, 5, 16, 17, 16, 17, 20, 21, 20, 21, 16, 17, 16, 17, 20, 21, 20, 21, 64, 65, 64, 65, 68, 69, 68, 69, 64, 65, 64, 65, 68, 69, 68

Offset: 0

Author

Antti Karttunen, Aug 03 2001

Keywords

• nonn
• base
• easy

Comments

a(n) is the formal derivative of x*n (evaluated at x=2 after being lifted to Z[x]) where n is interpreted as a polynomial in GF(2)[x] via its binary expansion. - Keith J. Bauer, Mar 17 2024

In the base 4 expansion of n, change 2 to 0 and 3 to 1. - Paolo Xausa, Feb 27 2025

Examples

			a(25) = 17 because 25 = 11001 in binary and when we AND this with 10101 we are left with 10001 = 17.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Ralf Stephan, Some divide-and-conquer sequences ...
• Ralf Stephan, Table of generating functions

Crossrefs

Cf. A004514, A063695 (remove even-positioned bits), A088442.

Programs

Haskell

a063694 0 = 0
a063694 n = 4 * a063694 n' + mod q 2
            where (n', q) = divMod n 4
-- Reinhard Zumkeller, Sep 26 2015

Magma

function A063694(n)
  if n le 1 then return n;
  else return 4*A063694(Floor(n/4)) + ((n mod 4) mod 2);
  end if; return A063694;
end function;
[A063694(n): n in [0..120]]; // G. C. Greubel, Dec 05 2022

Maple

every_other_pos := proc(nn, x, w) local n, i, s; n := nn; i := 0; s := 0; while(n > 0) do if((i mod 2) = w) then s := s + ((x^i)*(n mod x)); fi; n := floor(n/x); i := i+1; od; RETURN(s); end: [seq(every_other_pos(j, 2, 0), j=0..120)];

Mathematica

a[n_] := BitAnd[n, Sum[2^k, {k, 0, Log[2, n] // Floor, 2}]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 28 2016 *)
A063694[n_] := FromDigits[ReplaceAll[IntegerDigits[n, 4], {2 -> 0, 3 -> 1}], 4];
Array[A063694, 100, 0] (* Paolo Xausa, Feb 27 2025 *)

PARI

a(n)=sum(k=0,n,(-1)^k*2^k*floor(n/2^k))  /* since n> ceil(log(n)/log(2)) */

PARI

a(n)=if(n<0,0,sum(k=0,n,(-1)^k*2^k*floor(n/2^k))) /* since n> ceil(log(n)/log(2)) */

Python

def A063694(n): return n&((1<<(m:=n.bit_length())+(m&1))-1)//3 # Chai Wah Wu, Jan 30 2023

SageMath

def A063694(n):
    if (n<2): return n
    else: return 4*A063694(floor(n/4)) + ((n%4)%2)
[A063694(n) for n in range(121)] # G. C. Greubel, Dec 05 2022

Formula

a(n) = Sum_{k>=0} (-1)^k*2^k*floor(n/2^k).

a(n) + A063695(n) = n.

a(n) = n - 2*a(floor(n/2)). - Vladeta Jovovic, Feb 23 2003

G.f.: 1/(1-x) * Sum_{k>=0} (-2)^k*x^2^k/(1-x^2^k). - Ralf Stephan, May 05 2003

a(n) = 4*a(floor(n/4)) + (n mod 4) mod 2. - Reinhard Zumkeller, Sep 26 2015

a(n) = Sum_{k>=0} A030308(n,k)*A199572(k). - Philippe Deléham, Jan 12 2023