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

Author

Antti Karttunen, Apr 26 1999

Keywords

• nonn
• nice
• look
• easy
• base

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

Examples

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

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

Crossrefs

Cf. A048720, A048725, A048726, A048728.

Bisection of A003188 (even part).

See also A065620, A065621.

Cf. A242399.

Programs

Haskell

import Data.Bits (xor, shiftL)
a048724 n = n `xor` shiftL n 1 :: Integer
-- Reinhard Zumkeller, Mar 06 2013

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 *)

PARI

a(n)=bitxor(n,2*n) \\ Charles R Greathouse IV, Jan 04 2013

Python

def A048724(n): return n^(n<<1) # Chai Wah Wu, Apr 05 2021

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