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%I #75 Aug 08 2023 04:14:30
%S 1,0,3,2,5,4,7,6,9,8,11,10,13,12,15,14,17,16,19,18,21,20,23,22,25,24,
%T 27,26,29,28,31,30,33,32,35,34,37,36,39,38,41,40,43,42,45,44,47,46,49,
%U 48,51,50,53,52,55,54,57,56,59,58,61,60,63,62,65,64,67,66,69
%N Natural numbers, pairs reversed: a(n) = n + (-1)^n; also Nimsum n + 1.
%C A self-inverse permutation of the natural numbers.
%C Nonnegative numbers rearranged with least disturbance to maintain a(n) not equal to n. - _Amarnath Murthy_, Sep 13 2002
%C Essentially lodumo_2 of A059841. - _Philippe Deléham_, Apr 26 2009
%C a(n) = A180176(n) for n >= 20. - _Reinhard Zumkeller_, Aug 15 2010
%D E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 60.
%D J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.
%H Michael De Vlieger, <a href="/A004442/b004442.txt">Table of n, a(n) for n = 0..10000</a>
%H Paul Barry, <a href="https://arxiv.org/abs/2004.04577">On a Central Transform of Integer Sequences</a>, arXiv:2004.04577 [math.CO], 2020.
%H <a href="/index/Ni#Nimsums">Index entries for sequences related to Nim-sums</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F a(n) = n XOR 1. - _Odimar Fabeny_, Sep 05 2004
%F G.f.: (1-x+2x^2)/((1-x)*(1-x^2)). - Mitchell Harris, Jan 10 2005
%F a(n+1) = lod_2(A059841(n)). - _Philippe Deléham_, Apr 26 2009
%F a(n) = 2*n - a(n-1) - 1 with n > 0, a(0)=1. - _Vincenzo Librandi_, Nov 18 2010
%F a(n) = Sum_{k=1..n-1} (-1)^(n-1-k)*C(n+1,k). - _Mircea Merca_, Feb 07 2013
%F For n > 1, a(n)^a(n) == 1 (mod n). - _Thomas Ordowski_, Jan 04 2016
%F Sum_{n>=0,n<>1} (-1)^n/a(n) = log(2) = A002162. - _Peter McNair_, Aug 07 2023
%p a[0]:=1:a[1]:=0:for n from 2 to 70 do a[n]:=a[n-2]+2 od: seq(a[n], n=0..68); # _Zerinvary Lajos_, Feb 19 2008
%t Table[n + (-1)^n, {n, 0, 72}] (* or *)
%t CoefficientList[Series[(1 - x + 2x^2)/((1 - x)(1 - x^2)), {x, 0, 72}], x] (* _Robert G. Wilson v_, Jun 16 2006 *)
%t Flatten[Reverse/@Partition[Range[0,69],2]] (* or *) LinearRecurrence[{1,1,-1},{1,0,3},70] (* _Harvey P. Dale_, Jul 29 2018 *)
%o (Haskell)
%o import Data.List (transpose)
%o import Data.Bits (xor)
%o a004442 = xor 1 :: Integer -> Integer
%o a004442_list = concat $ transpose [a005408_list, a005843_list]
%o -- _Reinhard Zumkeller_, Jun 23 2013, Feb 01 2013, Oct 20 2011
%o (PARI) a(n)=n+(-1)^n \\ _Charles R Greathouse IV_, Nov 20 2012
%o (PARI) Vec((1-x+2*x^2)/((1-x)*(1-x^2)) + O(x^100)) \\ _Altug Alkan_, Feb 04 2016
%o (Python)
%o def a(n): return n^1
%o print([a(n) for n in range(69)]) # _Michael S. Branicky_, Jan 23 2022
%Y Cf. A003987, A004443, A004444. Equals A014681 - 1.
%Y Cf. A005843, A005408, A059841.
%Y Cf. A002162
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
%E Offset adjusted by _Reinhard Zumkeller_, Mar 05 2010
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