%I
%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 (a(n)1)*(a(n1)+1) = 2*A176222(n+1) for n>1; (a(n)1)*(a(n3)+1) = 2*A176222(n) for n>3.  _Bruno Berselli_, Nov 16 2010
%C For n >= 5, also the number of (undirected) Hamiltonian cycles in the (n2)Moebius ladder.  _Eric W. Weisstein_, May 06 2019
%C For n >= 4, also the number of (undirected) Hamiltonian cycles in the (n1)prism graph.  _Eric W. Weisstein_, May 06 2019
%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) = 2k1 = A005408(k), a(2k1) = 2k = A005843(k), k=1, 2, ...
%F O.g.f.: x*(x^2x+2)/[(x1)^2*(1+x)].  _R. J. Mathar_, Apr 06 2008
%F a(n) = n1+2*(n mod 2).  _Rolf Pleisch_, Apr 22 2008
%F a(n) = 2*na(n1)1 (with a(1)=2).  _Vincenzo Librandi_, Nov 16 2010
%F a(n) = n(1)^n. a(n)a(n1)a(n2)+a(n3) = 0 for n>3.  _Bruno Berselli_, Nov 16 2010
%t Table[{n + 1, n}, {n, 1, 100, 2}] // Flatten
%t Table[n  (1)^n, {n, 25}] (* _Eric W. Weisstein_, May 06 2019 *)
%o [ n eq 1 select 2 else Self(n1)+2*n1: 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)=n1+if(n%2,2) \\ _Charles R Greathouse IV_, Feb 24 2011
%Y Essentially the same as A014681.
%Y Odd numbers: A005408. Even numbers: A005843.
%Y Cf. A103889, A004442.
%K nonn,easy
%O 1,1
%A _Zak Seidov_, Feb 20 2005
