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 A065621 Reversing binary representation of n. Converting sum of powers of 2 in binary representation of a(n) to alternating sum gives n. 56
 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 (list; graph; refs; listen; history; text; internal format)
 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) = 2a(n), a(2n+1) = 2a(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) + (n mod 2) : a(n) := 4 * A003188(n div 2) + (n mod 2). - Matt Erbst (matt(AT)erbst.org), Jul 18 2006 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 CROSSREFS Cf. 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

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Last modified November 19 08:44 EST 2019. Contains 329318 sequences. (Running on oeis4.)