A063695 Remove even-positioned bits from the binary expansion of n.

0, 0, 2, 2, 0, 0, 2, 2, 8, 8, 10, 10, 8, 8, 10, 10, 0, 0, 2, 2, 0, 0, 2, 2, 8, 8, 10, 10, 8, 8, 10, 10, 32, 32, 34, 34, 32, 32, 34, 34, 40, 40, 42, 42, 40, 40, 42, 42, 32, 32, 34, 34, 32, 32, 34, 34, 40, 40, 42, 42, 40, 40, 42, 42, 0, 0, 2, 2, 0, 0, 2, 2, 8, 8, 10, 10, 8, 8, 10, 10, 0, 0

Offset: 0

Author

Antti Karttunen, Aug 03 2001

Keywords

• nonn
• base
• easy

Comments

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

Examples

			a(25) = 8 because 25 = 11001 in binary and when we AND this with 1010 we are left with 1000 = 8.
		

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 (bisection), A063694 (remove odd-positioned bits), A090569.

Programs

Haskell

a063695 0 = 0
a063695 n = 4 * a063695 n' + 2 * div q 2
            where (n', q) = divMod n 4
-- Reinhard Zumkeller, Sep 26 2015

Maple

[seq(every_other_pos(j,2,1),j=0..120)]; # Function every_other_pos given at A063694.

Mathematica

A063695[n_] := FromDigits[ReplaceAll[IntegerDigits[n, 4], {1 -> 0, 3 -> 2}], 4];
Array[A063695, 100, 0] (* Paolo Xausa, Feb 27 2025 *)

Python

def A063695(n): return n&((1<<(m:=n.bit_length())+(m&1^1))-1)//3 # Chai Wah Wu, Jan 30 2023

Formula

a(n) + A063694(n) = n.

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

From Ralf Stephan, Oct 06 2003: (Start)

G.f. 1/(1-x) * Sum_{k>=0} (-2)^k*2t^2/(1-t^2) where t = x^2^k.

Members of A004514 written twice.

(End)

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

a(n) = A090569(n+1)-1. - R. J. Mathar, Jun 22 2020

a(n) = 2*(n - A380110(n)). - Paolo Xausa, Feb 27 2025