

A048724


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


33



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
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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. AddisonWesley, Reading, MA, 1969, Vol. 2, p. 178, (exercise 4.1. Nr. 27)
H. D. Nguyen, A mixing of ProuhetThueMorse sequences and Rademacher functions, http://www.rowan.edu/colleges/csm/departments/math/facultystaff/nguyen/papers/mixingptmrademacher.pdf, 2014. See Example 20.  N. J. A. Sloane, May 24 2014


LINKS

T. D. Noe, Table of n, a(n) for n = 0..1023
R. Stephan, Some divideandconquer sequences ...
R. 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/(1x) * sum(k>=0, 2^k*(3tt^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.


MATHEMATICA

Table[ BitXor[2n, n], {n, 0, 65}] (from 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


CROSSREFS

Cf. A048720, A048725, A048726, A048728. Bisection of A003188.
See also A065620, A065621.
Cf. A242399.
Sequence in context: A246979 A246980 A095359 * A199126 A247569 A115389
Adjacent sequences: A048721 A048722 A048723 * A048725 A048726 A048727


KEYWORD

nonn,nice,easy


AUTHOR

Antti Karttunen, Apr 26, 1999


STATUS

approved



