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 A006068 a(n) is Gray-coded into n. (Formerly M2253) 146
 0, 1, 3, 2, 7, 6, 4, 5, 15, 14, 12, 13, 8, 9, 11, 10, 31, 30, 28, 29, 24, 25, 27, 26, 16, 17, 19, 18, 23, 22, 20, 21, 63, 62, 60, 61, 56, 57, 59, 58, 48, 49, 51, 50, 55, 54, 52, 53, 32, 33, 35, 34, 39, 38, 36, 37, 47, 46, 44, 45, 40, 41, 43, 42, 127, 126, 124, 125, 120, 121 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Equivalently, if binary expansion of n has m bits (say), compute derivative of n (A038554), getting sequence n' of length m-1; sort on n'. Inverse of sequence A003188 considered as a permutation of the nonnegative integers, i.e., a(A003188(n)) = n = A003188(a(n)). - Howard A. Landman, Sep 25 2001 The sequence exhibits glide reflections that grow fractally. These show up well on the scatterplot, also audibly using the "listen" link. - Peter Munn, Aug 18 2019 REFERENCES M. Gardner, Mathematical Games, Sci. Amer. Vol. 227 (No. 2, Feb. 1972), p. 107. M. Gardner, Knotted Doughnuts and Other Mathematical Entertainments. Freeman, NY, 1986, p. 15. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 0..1023 Joerg Arndt, Matters Computational (The Fxtbook), section 1.16 Gray code, C code inverse_gray_code() with "version 3" giving the PARI code here. Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020 Eric Rowland and Reem Yassawi, Profinite automata, arXiv:1403.7659 [math.DS], 2014. See p. 22. FORMULA a(n) = 2*a(ceiling((n+1)/2)) + A010060(n-1). If 3*2^(k-1) < n <= 2^(k+1), a(n) = 2^(k+1) - 1 - a(n-2^k); if 2^(k+1) < n <= 3*2^k, a(n) = a(n-2^k) + 2^k. - Henry Bottomley, Jan 10 2001 a(n) = n XOR [n/2] XOR [n/4] XOR [n/8] ... XOR [n/2^m] where m = [log(n)/log(2)] (for n>0) and [x] is integer floor of x. - Paul D. Hanna, Jun 04 2002 a(n) XOR [a(n)/2] = n. - Paul D. Hanna, Jan 18 2012 A066194(n) = a(n-1) + 1, n>=1. - Philippe Deléham, Apr 29 2005 a(n) = if n<2 then n else 2*m + (n mod 2 + m mod 2) mod 2, with m=a(floor(n/2)). - Reinhard Zumkeller, Aug 10 2010 a(n XOR m) = a(n) XOR a(m), where XOR is the bitwise exclusive-or operator, A003987. - Peter Munn, Dec 14 2019 a(0) = 0. For all n >= 0 if a(n) is even a(2*n) = 2*a(n), a(2*n+1) = 2*a(n)+1, else a(2*n) = 2*a(n)+1, a(2*n+1) = 2*a(n). - Yosu Yurramendi, Oct 12 2021 EXAMPLE The first few values of n' are -,-,1,0,10,11,01,00,100,101,111,110,010,011,001,000,... (for n=0..15) and to put these in lexicographic order we must take n in the order 0,1,3,2,7,6,4,5,15,14,12,13,8,9,11,10,... MAPLE a:= proc(n) option remember; `if`(n<2, n,       Bits[Xor](n, a(iquo(n, 2))))     end: seq(a(n), n=0..100);  # Alois P. Heinz, Apr 17 2018 MATHEMATICA a[n_] := BitXor @@ Table[Floor[n/2^m], {m, 0, Floor[Log[2, n]]}]; a[0]=0; Table[a[n], {n, 0, 69}] (* Jean-François Alcover, Jul 19 2012, after Paul D. Hanna *) Table[Fold[BitXor, n, Quotient[n, 2^Range[BitLength[n] - 1]]], {n, 0, 70}] (* Jan Mangaldan, Mar 20 2013 *) PROG (PARI) {a(n)=local(B=n); for(k=1, floor(log(n+1)/log(2)), B=bitxor(B, n\2^k)); B} /* Paul D. Hanna, Jan 18 2012 */ (PARI) /* the following routine needs only O(log_2(n)) operations */ a(n)= {     my( s=1, ns );     while ( 1,         ns = n >> s;         if ( 0==ns, break() );         n = bitxor(n, ns);         s <<= 1;     );     return ( n ); } /* Joerg Arndt, Jul 19 2012 */ (Haskell) a006068 n = foldl xor 0 \$                   map (div n) \$ takeWhile (<= n) a000079_list :: Integer -- Reinhard Zumkeller, Apr 28 2012 (Python) def a(n):     s=1     while True:         ns=n>>s         if ns==0: break         n=n^ns         s<<=1     return n print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 07 2017, after PARI code by Joerg Arndt (R) nmax <- 63 # by choice a <- vector() for(n in 1:nmax){   ones <- which(as.integer(intToBits(n)) == 1)   nbit <- as.integer(intToBits(n))[1:tail(ones, n = 1)]   level <- 0; anbit <- nbit; anbit.s <- nbit   while(sum(anbit.s) > 0){     s <- 2^level; if(s > length(anbit.s)) break     anbit.s <- c(anbit[-(1:s)], rep(0, s))     anbit <- bitwXor(anbit, anbit.s)     level <- level + 1   }   a <- c(a, sum(anbit*2^(0:(length(anbit) - 1)))) } (a <- c(0, a)) # Yosu Yurramendi, Oct 12 2021, after PARI code by Joerg Arndt CROSSREFS Cf. A038554, A005811, A003188, A014550, A003100. Cf. A054429, A180200. - Reinhard Zumkeller, Aug 15 2010 Cf. A000079, A055975 (first differences), A209281 (binary weight). A003987, A010060 are used to express relationship between terms of this sequence. Sequence in context: A304083 A276441 A153141 * A154436 A269402 A268934 Adjacent sequences:  A006065 A006066 A006067 * A006069 A006070 A006071 KEYWORD nonn,easy,nice,look,hear,changed AUTHOR EXTENSIONS More terms from Henry Bottomley, Jan 10 2001 STATUS approved

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Last modified May 23 23:07 EDT 2022. Contains 353993 sequences. (Running on oeis4.)