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A permutation of the integers (a fractal sequence): a(n) = A006068(n-1) + 1.
7

%I #52 Jul 01 2022 17:34:23

%S 1,2,4,3,8,7,5,6,16,15,13,14,9,10,12,11,32,31,29,30,25,26,28,27,17,18,

%T 20,19,24,23,21,22,64,63,61,62,57,58,60,59,49,50,52,51,56,55,53,54,33,

%U 34,36,35,40,39,37,38,48,47,45,46,41,42,44,43,128,127,125,126,121,122

%N A permutation of the integers (a fractal sequence): a(n) = A006068(n-1) + 1.

%C With an initial zero, inverse of the Gray Code (A003188). See also A006068. - _Robert G. Wilson v_, Jun 22 2014

%C I suspect the above comment refers to function A105081(n) = 1 + A003188(n - 1), n >= 1. - _Antti Karttunen_, Feb 15 2016

%H Antti Karttunen, <a href="/A066194/b066194.txt">Table of n, a(n) for n = 1..8192</a> (first 1024 terms from Robert G. Wilson v)

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F a(n) = A006068(n-1) + 1, n >= 1. - _Philippe Deléham_, Apr 29 2005

%F a(n) = A006068(A268717(n)), composition of related permutations. - _Antti Karttunen_, Feb 14 2016

%F a(n) = 1 + Sum_{j=1..n-1} (1/6)*(-3 + (-1)^A007814(j) + 2^(A007814(j) + 3))*(-1)^(A000120(j) + 1). - _John Erickson_, Oct 18 2018

%e Third nesting gives {1,2,4,3, 8,7,5,6} by means of joining the lists {1,2,4,3} = second nesting and {8,7,6,5} permuted by {1,2,4,3} giving {8,7,5,6}.

%t Nest[ Join[ #, (Length[ #] + Range[ Length[ #], 1, -1 ])[[ # ]]] &, {1}, 7 ]

%t GrayCode[n_] := BitXor[n, Floor[n/2]]; t = Array[ GrayCode, 1000, 0]; Table[ Position[ t, n], {n, 0, 100}] // Flatten (* _Robert G. Wilson v_, Jun 22 2014 *)

%o (Scheme) (define (A066194 n) (+ 1 (A006068 (- n 1)))) ;; _Antti Karttunen_, Feb 14 2016

%o (Python)

%o def A066194(n):

%o k, m = n-1, n-1>>1

%o while m > 0:

%o k ^= m

%o m >>= 1

%o return k+1 # _Chai Wah Wu_, Jul 01 2022

%Y Inverse: A105081.

%Y Cf. A003188, A006068, A268717.

%K easy,nonn

%O 1,2

%A _Wouter Meeussen_, Dec 15 2001

%E Deléham's formula added to the name by _Antti Karttunen_, Feb 14 2016