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%I #18 Aug 28 2016 18:23:42
%S 0,2,1,4,6,3,8,5,10,12,7,14,9,16,11,18,20,13,22,15,24,17,26,19,28,30,
%T 21,32,23,34,25,36,27,38,29,40,42,31,44,33,46,35,48,37,50,39,52,41,54,
%U 56,43,58,45,60,47,62,49,64,51,66,53,68,55,70,72,57,74,59,76,61,78,63,80
%N Let m = floor(sqrt(n)); if n and m have the same parity, a(n) = n + m, otherwise a(n) = n - m.
%C This is a permutation of the nonnegative integers; it can also be generated by the rule (with m = floor(sqrt(n))): if n - m is not yet in the sequence, a(n) = n - m, otherwise a(n) = n + m. All cycles in this permutation are finite. There is one relatively large cycle starting at n = 4k^2 - 2k + 1 for each k and k 2-cycles for n = (2k - 1)^2 + 2i and (2k - 1)^2 + 2k - 1 + 2i with 0 <= i < k.
%C a(A133280(n,k)) mod 2 = 0 and a(A195437(n,k)) mod 2 = 1, 0 <= k < n. [_Reinhard Zumkeller_, Oct 12 2011]
%H T. D. Noe, <a href="/A127366/b127366.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%t a[n_] := If[m = Floor[Sqrt[n]]; OddQ[n] && OddQ[m] || EvenQ[n] && EvenQ[m], n+m, n-m]; Table[ a[n], {n, 0, 72}](* _Jean-François Alcover_, Nov 30 2011 *)
%o (Haskell)
%o a127366 n | even n' = n'
%o | otherwise = 2*n - n'
%o where n' = n + a000196 n
%o -- _Reinhard Zumkeller_, Oct 12 2011
%Y Cf. A127367.
%Y Cf. A000196.
%K nice,nonn
%O 0,2
%A _Franklin T. Adams-Watters_, Jan 11 2007
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