A065621 Reversing binary representation of n. Converting sum of powers of 2 in binary representation of a(n) to alternating sum gives n.

1, 2, 7, 4, 13, 14, 11, 8, 25, 26, 31, 28, 21, 22, 19, 16, 49, 50, 55, 52, 61, 62, 59, 56, 41, 42, 47, 44, 37, 38, 35, 32, 97, 98, 103, 100, 109, 110, 107, 104, 121, 122, 127, 124, 117, 118, 115, 112, 81, 82, 87, 84, 93, 94, 91, 88, 73, 74, 79, 76, 69, 70, 67, 64, 193

Offset

1,2

Comments

a(0)=0. The alternation is applied only to the nonzero bits and does not depend on the exponent of two. All integers have a unique reversing binary representation (see cited exercise for proof). Complement of A048724.

A permutation of the "odious" numbers A000069.

Write n-1 and 2n-1 in binary and add them mod 2; example: n = 6, n-1 = 5 = 101 in binary, 2n-1 = 11 = 1011 in binary and their sum is 1110 = 14, so a(6) = 14. - Philippe Deléham, Apr 29 2005

As already pointed out, this is a permutation of the odious numbers A000069 and A010060(A000069(n)) = 1, so A010060(a(n)) = 1; and A010060(A048724(n)) = 0. - Philippe Deléham, Apr 29 2005. Also a(n) = A000069(A003188(n - 1)).

References

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

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..8192

Formula

a(n) = if n=0 or n=1 then n else b+2*a(b+(1-2*b)*n)/2) where b is the least significant bit in n.

a(n) = n XOR 2 (n - (n AND -n)).

a(1) = 1, a(2n) = 2*a(n), a(2n+1) = 2*a(n+1) - 2(-1)^n + 1. - Ralf Stephan, Aug 20 2003

a(n) = A048724(n-1) - (-1)^n. - Ralf Stephan, Sep 10 2003

a(n) = Sum_{k=0..n} (1-(-1)^round(-n/2^k))/2*2^k. - Benoit Cloitre, Apr 27 2005

Closely related to Gray codes in another way: a(n) = 2 * A003188(n-1) + (n mod 2); a(n) = 4 * A003188((n-1) div 2) + (n mod 4). - Matt Erbst (matt(AT)erbst.org), Jul 18 2006 [corrected by Peter Munn, Jan 30 2021]

a(n) = n XOR 2(n AND NOT -n). - Chai Wah Wu, Jun 29 2022

a(n) = A003188(2n-1). - Friedjof Tellkamp, Jan 18 2024

Example

a(5) = 13 = 8 + 4 + 1 -> 8 - 4 + 1 = 5.

Mathematica

f[n_] := BitXor[n, 2 n + 1]; Array[f, 60, 0] (* Robert G. Wilson v, Jun 09 2010 *)

Prog

(PARI) a(n)=if(n<2, 1, if(n%2==0, 2*a(n/2), 2*a((n+1)/2)-2*(-1)^((n-1)/2)+1))
(Haskell)
import Data.Bits (xor, (.&.))
a065621 n = n `xor` 2 * (n - n .&. negate n) :: Integer
-- Reinhard Zumkeller, Mar 26 2014
(Python)
def a(n): return n^(2*(n - (n & -n))) # Indranil Ghosh, Jun 04 2017
(Python)
def A065621(n): return n^ (n&~-n)<<1 # Chai Wah Wu, Jun 29 2022

Crossrefs

Cf. A003188, A065620, A048724, A072219, A073122.

Differs from A115857 for the first time at n=19, where a(19)=55, while A115857(19)=23. Cf. A104895, A115872, A114389, A114390, A105081.

Cf. A245471.

Sequence in context: A329064 A102514 A115857 * A036565 A054787 A190716

Adjacent sequences: A065618 A065619 A065620 * A065622 A065623 A065624

Keyword

easy,nonn,look

Author

Marc LeBrun, Nov 07 2001

Extensions

More terms from Ralf Stephan, Sep 08 2003

Status

approved