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%I #22 Dec 25 2023 01:51:58
%S 2,1,0,3,6,5,4,7,10,9,8,11,14,13,12,15,18,17,16,19,22,21,20,23,26,25,
%T 24,27,30,29,28,31,34,33,32,35,38,37,36,39,42,41,40,43,46,45,44,47,50,
%U 49,48,51,54,53,52,55,58,57,56,59,62,61,60,63,66,65,64
%N Permutation of the nonnegative integers: [4k+2, 4k+1, 4k, 4k+3, ...].
%C Fixed points are the odd numbers (A005408).
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-1).
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F G.f.: (2 - 3*x + 2*x^2 + x^3)/((x - 1)^2*(1 + x^2)).
%F a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>3.
%F a(n) = n + (1 + (-1)^n)*(-1)^(n*(n+1)/2).
%F a(n) = 4*floor((n+1)/4) - (n mod 4) + 2.
%F a(n) = A092486(n) - 1.
%F a(n) = n + A176742(n) for n>0.
%F a(2n) = 2*A004442(n), a(2n+1) = A005408(n).
%F a(-n-1) = -A263449(n).
%F a(n+1) = a(n) - A132429(n+1)*(-1)^n.
%F Sum_{n>=0, n!=2} (-1)^(n+1)/a(n) = 1 - log(2) (A244009). - _Amiram Eldar_, Dec 25 2023
%p A263426:=n->n + (1 + (-1)^n)*(-1)^(n*(n + 1)/2): seq(A263426(n), n=0..80);
%t Table[n + (1 + (-1)^n)*(-1)^(n*(n + 1)/2), {n, 0, 80}]
%o (Magma) [n+(1+(-1)^n)*(-1)^(n*(n+1) div 2) : n in [0..80]];
%o (Magma) /* By definition: */ &cat[[4*k+2,4*k+1,4*k,4*k+3]: k in [0..20]]; // _Bruno Berselli_, Nov 08 2015
%o (PARI) Vec((2-3*x+2*x^2+x^3)/((x-1)^2*(1+x^2)) + O(x^100)) \\ _Altug Alkan_, Oct 19 2015
%Y Cf. A004442, A005408, A092486, A132429, A176742, A244009, A263449.
%K nonn,easy
%O 0,1
%A _Wesley Ivan Hurt_, Oct 17 2015
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