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%I #63 May 04 2023 06:32:23
%S 2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15,18,17,20,19,22,21,24,23,26,25,
%T 28,27,30,29,32,31,34,33,36,35,38,37,40,39,42,41,44,43,46,45,48,47,50,
%U 49,52,51,54,53,56,55,58,57,60,59,62,61,64,63,66,65,68,67,70,69,72,71
%N Odd and even positive integers swapped.
%C For n >= 5, also the number of (undirected) Hamiltonian cycles in the (n-2)-Moebius ladder. - _Eric W. Weisstein_, May 06 2019
%C For n >= 4, also the number of (undirected) Hamiltonian cycles in the (n-1)-prism graph. - _Eric W. Weisstein_, May 06 2019
%C The lexicographically first involution of the natural numbers with no fixed points. - _Alexander Fraebel_, Sep 08 2020
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HamiltonianCycle.html">Hamiltonian Cycle</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MoebiusLadder.html">Moebius Ladder</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrismGraph.html">Prism Graph</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F a(2k) = 2k-1 = A005408(k), a(2k-1) = 2k = A005843(k), k=1, 2, ...
%F O.g.f.: x*(x^2-x+2)/((x-1)^2*(1+x)). - _R. J. Mathar_, Apr 06 2008
%F a(n) = n-1+2*(n mod 2). - _Rolf Pleisch_, Apr 22 2008
%F a(n) = 2*n-a(n-1)-1 (with a(1)=2). - _Vincenzo Librandi_, Nov 16 2010
%F From _Bruno Berselli_, Nov 16 2010: (Start)
%F a(n) = n - (-1)^n.
%F a(n) - a(n-1) - a(n-2) + a(n-3) = 0 for n > 3.
%F (a(n) - 1)*(a(n-1) + 1) = 2*A176222(n+1) for n > 1.
%F (a(n) - 1)*(a(n-3) + 1) = 2*A176222(n) for n > 3. (End)
%F E.g.f.: 1 - exp(-x) + x*exp(x). - _Stefano Spezia_, May 03 2023
%t Table[{n + 1, n}, {n, 1, 100, 2}] // Flatten
%t Table[n - (-1)^n, {n, 25}] (* _Eric W. Weisstein_, May 06 2019 *)
%o (Magma) [n eq 1 select 2 else -Self(n-1)+2*n-1: n in [1..72]];
%o (Haskell)
%o import Data.List (transpose)
%o a103889 n = n - 1 + 2 * mod n 2
%o a103889_list = concat $ transpose [tail a005843_list, a005408_list]
%o -- _Reinhard Zumkeller_, Jun 23 2013, Feb 21 2011
%o (PARI) a(n)=n-1+if(n%2,2) \\ _Charles R Greathouse IV_, Feb 24 2011
%o (Python)
%o def a(n): return n+1 if n&1 else n-1
%o print([a(n) for n in range(1, 73)]) # _Michael S. Branicky_, May 03 2023
%Y Essentially the same as A014681.
%Y Odd numbers: A005408. Even numbers: A005843.
%Y Cf. A004442.
%K nonn,easy
%O 1,1
%A _Zak Seidov_, Feb 20 2005
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