A048724 Write n and 2n in binary and add them mod 2.

0, 3, 6, 5, 12, 15, 10, 9, 24, 27, 30, 29, 20, 23, 18, 17, 48, 51, 54, 53, 60, 63, 58, 57, 40, 43, 46, 45, 36, 39, 34, 33, 96, 99, 102, 101, 108, 111, 106, 105, 120, 123, 126, 125, 116, 119, 114, 113, 80, 83, 86, 85, 92, 95, 90, 89, 72, 75, 78, 77, 68, 71, 66, 65, 192

Offset

0,2

Comments

Reversing binary representation of -n. Converting sum of powers of 2 in binary representation of a(n) to alternating sum gives -n. Note that 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 A065621. - Marc LeBrun, Nov 07 2001

A permutation of the "evil" numbers A001969. - Marc LeBrun, Nov 07 2001

A048725(n) = a(a(n)). - Reinhard Zumkeller, Nov 12 2004

References

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

Links

T. D. Noe, Table of n, a(n) for n = 0..1023

P. Mathonet, M. Rigo, M. Stipulanti and N. Zénaïdi, On digital sequences associated with Pascal's triangle, arXiv:2201.06636 [math.NT], 2022.

H. D. Nguyen, A mixing of Prouhet-Thue-Morse sequences and Rademacher functions, 2014. See Example 20. - N. J. A. Sloane, May 24 2014

Ralf Stephan, Some divide-and-conquer sequences ...

Ralf Stephan, Table of generating functions

Formula

a(n) = Xmult(n, 3) (or n XOR (n<<1)).

a(n) = A065621(-n).

a(2n) = 2a(n), a(2n+1) = 2a(n) + 2(-1)^n + 1.

G.f. 1/(1-x) * sum(k>=0, 2^k*(3t-t^3)/(1+t)/(1+t^2), t=x^2^k). - Ralf Stephan, Sep 08 2003

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

a(n) = A001969(A003188(n)). - Philippe Deléham, Apr 29 2005

a(n) = A106409(2*n) for n>0. - Reinhard Zumkeller, May 02 2005

a(n) = A142149(2*n). - Reinhard Zumkeller, Jul 15 2008

Example

12 = 1100 in binary, 24=11000 and their sum is 10100=20, so a(12)=20.
a(4) = 12 = + 8 + 4 -> - 8 + 4 = -4.

Maple

a:= n-> Bits[Xor](n, n+n):
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 06 2016

Mathematica

Table[ BitXor[2n, n], {n, 0, 65}] (* Robert G. Wilson v, Jul 06 2006 *)

Prog

(PARI) a(n)=bitxor(n, 2*n) \\ Charles R Greathouse IV, Jan 04 2013
(Haskell)
import Data.Bits (xor, shiftL)
a048724 n = n `xor` shiftL n 1 :: Integer
-- Reinhard Zumkeller, Mar 06 2013
(Python)
def A048724(n): return n^(n<<1) # Chai Wah Wu, Apr 05 2021

Crossrefs

Cf. A048720, A048725, A048726, A048728.

Bisection of A003188.

See also A065620, A065621.

Cf. A242399.

Sequence in context: A246979 A246980 A095359 * A292682 A334748 A334079

Adjacent sequences: A048721 A048722 A048723 * A048725 A048726 A048727

Keyword

nonn,nice,look,easy,base

Author

Antti Karttunen, Apr 26 1999

Status

approved