A268717 Permutation of natural numbers: a(0) = 0, a(n) = A003188(1+A006068(n-1)), where A003188 is binary Gray code and A006068 is its inverse.

0, 1, 3, 6, 2, 12, 4, 7, 5, 24, 8, 11, 9, 13, 15, 10, 14, 48, 16, 19, 17, 21, 23, 18, 22, 25, 27, 30, 26, 20, 28, 31, 29, 96, 32, 35, 33, 37, 39, 34, 38, 41, 43, 46, 42, 36, 44, 47, 45, 49, 51, 54, 50, 60, 52, 55, 53, 40, 56, 59, 57, 61, 63, 58, 62, 192, 64, 67, 65, 69, 71, 66, 70, 73, 75, 78, 74, 68, 76, 79, 77, 81

Offset

0,3

Links

Antti Karttunen, Table of n, a(n) for n = 0..8191

Index entries for sequences that are permutations of the natural numbers

Formula

a(n) = A003188(A066194(n)) = A003188(1+A006068(n-1)).

Other identities. For all n >= 0:

A101080(n,a(n+1)) = 1. [The Hamming distance between n and a(n+1) is always one.]

A268726(n) = A000523(A003987(n, a(n+1))). [A268726 gives the index of the toggled bit.]

Mathematica

A003188[n_] := BitXor[n, Floor[n/2]]; A006068[n_] := If[n == 0, 0, BitXor @@ Table[Floor[n/2^m], {m, 0, Floor[Log[2, n]]}]]; a[n_] := If[n == 0, 0, A003188[1 + A006068[n-1]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 23 2016 *)

Prog

(Scheme) (define (A268717 n) (if (zero? n) n (A003188 (A066194 n))))
(PARI) A003188(n) = bitxor(n, floor(n/2));
A006068(n) = if(n<2, n, {my(m = A006068(floor(n/2))); 2*m + (n%2 + m%2)%2});
for(n=0, 100, print1(if(n<1, 0, A003188(1 + A006068(n - 1)))", ")) \\ Indranil Ghosh, Mar 31 2017
(Python)
def A003188(n): return n^(n//2)
def A006068(n):
    if n<2: return n
    m = A006068(n//2)
    return 2*m + (n%2 + m%2)%2
def a(n): return 0 if n<1 else A003188(1 + A006068(n - 1))
print([a(n) for n in range(0, 101)]) # Indranil Ghosh, Mar 31 2017
(Python)
def A268717(n):
    k, m = n-1, n-1>>1
    while m > 0:
        k ^= m
        m >>= 1
    return k+1^ k+1>>1 # Chai Wah Wu, Jun 29 2022

Crossrefs

Inverse: A268718.

Cf. A000523, A003188, A003987, A006068, A066194, A101080, A268726, A268727.

Row 1 and column 1 of array A268715 (without the initial zero).

Row 1 of array A268820.

Cf. A092246 (fixed points).

Cf. A268817 ("square" of this permutation).

Cf. A268821 ("shifted square"), A268823 ("shifted cube") and also A268825, A268827 and A268831 ("shifted higher powers").

Sequence in context: A257506 A210035 A210199 * A002516 A073807 A345671

Adjacent sequences: A268714 A268715 A268716 * A268718 A268719 A268720

Keyword

nonn

Author

Antti Karttunen, Feb 12 2016

Status

approved