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Permutation of nonnegative integers: a(0) = 0, a(n) = A268717(1 + A268821(n-1)).
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%I #22 Mar 11 2024 18:05:25

%S 0,1,3,2,6,5,7,15,13,4,12,27,25,8,24,14,10,9,11,51,49,16,48,22,18,17,

%T 19,26,30,29,31,23,21,28,20,99,97,32,96,38,34,33,35,42,46,45,47,39,37,

%U 44,36,50,54,53,55,63,61,52,60,43,41,56,40,62,58,57,59,195,193,64,192,70,66,65,67,74,78,77,79,71,69,76,68,82,86,85

%N Permutation of nonnegative integers: a(0) = 0, a(n) = A268717(1 + A268821(n-1)).

%C The "third shifted power" of permutation A268717.

%H Antti Karttunen, <a href="/A268823/b268823.txt">Table of n, a(n) for n = 0..8191</a>

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

%F a(0), for n >= 1, a(n) = A268717(1 + A268821(n-1)).

%F a(0) = 0, a(1) = 1, and for n > 1, a(n) = A268717(1 + A268717(1 + A268717(n-2))).

%F For n >= 3, a(n) = A003188(3+A006068(n-3)). - _Antti Karttunen_, Mar 11 2024

%o (Scheme) (define (A268823 n) (if (<= n 1) n (A268717 (+ 1 (A268717 (+ 1 (A268717 (- n 2))))))))

%o (PARI) A003188(n) = bitxor(n, floor(n/2));

%o A006068(n) = if(n<2, n, {my(m = A006068(floor(n/2))); 2*m + (n%2 + m%2)%2});

%o A268717(n) = if(n<1, 0, A003188(1 + A006068(n - 1)));

%o for(n=0, 100, print1(if(n<2, n, A268717(1 + A268717(1 + A268717(n - 2)))),", ")) \\ _Indranil Ghosh_, Mar 31 2017

%o (Python)

%o def A003188(n): return n^(n//2)

%o def A006068(n):

%o if n<2: return n

%o else:

%o m=A006068(n//2)

%o return 2*m + (n%2 + m%2)%2

%o def A268717(n): return 0 if n<1 else A003188(1 + A006068(n - 1))

%o def a(n): return A268717(1 + A268717(1 + A268717(n - 2))) if n>1 else n

%o print([a(n) for n in range(101)]) # _Indranil Ghosh_, Mar 31 2017

%Y Inverse: A268824.

%Y Cf. A003188, A006068, A268717, A268821, A268825, A268676.

%Y Row 3 of array A268820.

%K nonn

%O 0,3

%A _Antti Karttunen_, Feb 14 2016