A065620 a(0)=0; thereafter a(2n) = 2a(n), a(2n+1) = -2a(n) + 1.

0, 1, 2, -1, 4, -3, -2, 3, 8, -7, -6, 7, -4, 5, 6, -5, 16, -15, -14, 15, -12, 13, 14, -13, -8, 9, 10, -9, 12, -11, -10, 11, 32, -31, -30, 31, -28, 29, 30, -29, -24, 25, 26, -25, 28, -27, -26, 27, -16, 17, 18, -17, 20, -19, -18, 19, 24, -23, -22, 23, -20, 21, 22, -21, 64, -63, -62, 63, -60, 61, 62, -61, -56

Offset: 0

Author

Marc LeBrun, Nov 07 2001

Keywords

• base
• easy
• sign
• look

Comments

Reversing binary value of n: convert sum of powers of 2 in binary representation of n to alternating sum.

The alternation is applied only to the nonzero bits and does not depend on the exponent of 2. All integers have a unique reversing binary representation (see cited Knuth exercise for proof).

Examples

			11 = 1 + 2 + 8 -> 1 - 2 + 8 = 7 = a(11).
		

References

• Donald E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 178 (exercise 4.1. Nr. 27).

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..20000 (First 8192 terms from Reinhard Zumkeller)

Crossrefs

Cf. A010060, A048724, A065621, A246159, A246160, A265263.

The negative of entry A104895.

Programs

Haskell

import Data.List (transpose)
a065620 n = a065620_list !! n
a065620_list = 1 : concat (transpose [zs, map ((+ 1) . negate) zs])
               where zs = map (* 2) a065620_list
-- Reinhard Zumkeller, Mar 26 2014

Maple

f:=proc(n) option remember; if n=0 then 0 elif (n mod 2) = 0 then 2*f(n/2) else 1-2*f((n-1)/2); fi; end;
[seq(f(n),n=0..100)]; # N. J. A. Sloane, Apr 25 2018

Mathematica

a[0] = 0; a[n_]:= a[n]= If[OddQ[n], 1 - 2*a[(n-1)/2], 2*a[n/2]]; Array[a, 100, 0] (* Amiram Eldar, Sep 05 2023 *)

PARI

A065620(n,c=1)=sum(i=0,logint(n+!n,2),if(bittest(n,i),(-1)^c++<M. F. Hasler, Apr 16 2018

Python

def a(n): return n if n<3 else 2*a(n/2) if n%2==0 else -2*a((n - 1)/2) + 1 # Indranil Ghosh, Jun 07 2017

Python

def A065620(n):
    c, a, b = 0, -1, 1
    for j in bin(n)[-1:1:-1]:
        if int(j):
            c += (a:=-a)*b
        b <<= 1
    return c # Chai Wah Wu, Sep 19 2023

Scheme

(definec (A065620 n) (cond ((zero? n) n) ((even? n) (* 2 (A065620 (/ n 2)))) (else (+ 1 (* -2 (A065620 (/ (- n 1) 2)))))))
;; using memoization-macro definec from IntSeq-library of Antti Karttunen, Aug 18 2014

Formula

a(n) = if n<2 then n else b+2*(1-2*b)*a((n-b)/2) where b is the least significant bit in n.

From Antti Karttunen, Aug 18 2014: (Start)

a(n) = A246160(n) - A246159(n).

a(A065621(n)) = n.

a(A048724(n)) = -n. (End)

a(n) = -A104895(n). - M. F. Hasler, Apr 16 2018

a(n) = (2*A265263(n) - n) * (2*A010060(n) - 1). - Alan Michael Gómez Calderón, Jul 01 2025

Extensions

Formula fixed by Reinhard Zumkeller, Mar 26 2014

Extended to a(0) = 0 by M. F. Hasler, Apr 16 2018

Edited by N. J. A. Sloane, Apr 25 2018