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
Conjecture: a(n) = a(A053645(A063946(n))) + A053644(n) for n > 0 with a(0) = 0. - Mikhail Kurkov, Sep 09 2023
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
AUTHOR
EXTENSIONS
More terms from Henry Bottomley, Jan 10 2001
STATUS
approved