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%I #30 Dec 25 2023 01:51:54
%S 3,0,1,2,7,4,5,6,11,8,9,10,15,12,13,14,19,16,17,18,23,20,21,22,27,24,
%T 25,26,31,28,29,30,35,32,33,34,39,36,37,38,43,40,41,42,47,44,45,46,51,
%U 48,49,50,55,52,53,54,59,56,57,58,63,60,61,62,67,64,65
%N Permutation of the nonnegative integers: [4k+3, 4k, 4k+1, 4k+2, ...].
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F G.f.: (3-3*x+x^2+x^3+2*x^4)/((x-1)^2*(1+x+x^2+x^3)).
%F a(n) = a(n-1)+a(n-4)-a(n-5) for n>4.
%F a(n) = n+1-2*(-1)^((n+1)*(n+2)*(n+3)/2).
%F a(2n) = A166519(n), a(2n+1) = A005843(n).
%F a(n) = n+(-1)^n+2*cos(n*Pi/2). - _Wesley Ivan Hurt_, May 09 2021
%F Sum_{n>=0, n!=1} (-1)^n/a(n) = log(2) (A002162). - _Amiram Eldar_, Dec 25 2023
%p A263313:=n->n+1-2*(-1)^((n+1)*(n+2)*(n+3)/2): seq(A263313(n), n=0..100);
%t Table[n + 1 - 2*(-1)^((n + 1)*(n + 2)*(n + 3)/2), {n, 0, 100}] (* or *) CoefficientList[Series[(3 - 3*x + x^2 + x^3 + 2*x^4)/((x - 1)^2*(1 + x + x^2 + x^3)), {x, 0, 100}], x]
%t Flatten[RotateRight/@Partition[Range[0,100],4]] (* or *) LinearRecurrence[ {1,0,0,1,-1},{3,0,1,2,7},100] (* _Harvey P. Dale_, Jul 01 2019 *)
%o (Magma) [n+1-2*(-1)^((n+1)*(n+2)*(n+3) div 2) : n in [0..100]];
%o (PARI) Vec((3-3*x+x^2+x^3+2*x^4)/((x-1)^2*(1+x+x^2+x^3)) + O(x^100)) \\ _Altug Alkan_, Oct 19 2015
%Y Cf. A002162, A005843, A166519.
%K nonn,easy
%O 0,1
%A _Wesley Ivan Hurt_, Oct 19 2015
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