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A284307 Permutation of the natural numbers partitioned into quadruples [4k-3, 4k, 4k-2, 4k-1], k > 0.

%I A284307 #44 Dec 12 2023 07:47:04
%S A284307 1,4,2,3,5,8,6,7,9,12,10,11,13,16,14,15,17,20,18,19,21,24,22,23,25,28,
%T A284307 26,27,29,32,30,31,33,36,34,35,37,40,38,39,41,44,42,43,45,48,46,47,49,
%U A284307 52,50,51,53,56,54,55,57,60,58,59,61,64,62,63,65,68,66,67
%N A284307 Permutation of the natural numbers partitioned into quadruples [4k-3, 4k, 4k-2, 4k-1], k > 0.
%C A284307 Partition the natural number sequence into quadruples starting with (1, 2, 3, 4); swap the third and fourth element, then swap the second and third element; repeat for all quadruples.
%H A284307 Guenther Schrack, <a href="/A284307/b284307.txt">Table of n, a(n) for n = 1..10000</a>
%H A284307 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%H A284307 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A284307 a(1)=1, a(2)=4, a(3)=2, a(4)=3, a(n) = a(n-4) + 4, n > 4.
%F A284307 O.g.f.: (x^4 + x^3 - 2*x^2 + 3x - 1)/(x^5 - x^4 - x + 1).
%F A284307 a(n) = n + ((-1)^n - (-1)^(n*(n-1)/2)*(1 + 2*(-1)^n))/2.
%F A284307 a(n) = n + (-1)^n*(1 - (-1)^(n*(n-1)/2) - (i^n - (-i)^n))/2.
%F A284307 Linear recurrence: a(n) = a(n-1) + a(n-4) - a(n-5), n > 5.
%F A284307 First differences, periodic: (3, -2, 1, 2), repeat.
%F A284307 a(n) = (2*n - 3*cos(n*Pi/2) + cos(n*Pi) + sin(n*Pi/2))/2. - _Wesley Ivan Hurt_, Apr 01 2017
%t A284307 Table[n + ((-1)^n - (-1)^(n (n - 1)/2) (1 + 2 (-1)^n))/2, {n, 68}] (* _Michael De Vlieger_, Mar 28 2017 *)
%t A284307 LinearRecurrence[{1,0,0,1,-1},{1,4,2,3,5},70] (* or *) {#[[1]],#[[4]], #[[2]],#[[3]]}&/@Partition[Range[70],4]//Flatten(* _Harvey P. Dale_, Sep 27 2017 *)
%o A284307 (MATLAB) a = [1 4 2 3];
%o A284307 max = (specify);
%o A284307 for n = 5:max
%o A284307    a(n) = a(n-4) + 4;
%o A284307 end;
%o A284307 (PARI) for(n=1, 68, print1(n + ((-1)^n - (-1)^(n*(n - 1)/2)*(1 + 2*(-1)^n))/2,", ")) \\ _Indranil Ghosh_, Mar 29 2017
%Y A284307 Inverse: A056699.
%Y A284307 Subsequences:
%Y A284307   elements with odd index: A042963(n), n > 0
%Y A284307   elements with even index: A014601(A103889(n)), n > 0
%Y A284307   odd elements: A005408(n-1), n > 0
%Y A284307   indices of odd elements: A042948(n), n > 0
%Y A284307   even elements: 2*A103889(n), n > 0
%Y A284307   indices of even elements: A042964(n), n > 0
%Y A284307 Sequence of fixed points: A016813(n-1), n > 0
%Y A284307 Every fourth element starting at:
%Y A284307   n=1: a(4n-3) = 4n-3 = A016813(n-1), n > 0
%Y A284307   n=2: a(4n-2) = 4n = A008586(n), n > 0
%Y A284307   n=3: a(4n-1) = 4n-2 = A016825(n-1), n > 0
%Y A284307   n=4: a(4n) = 4n-1 = A004767(n-1), n > 0
%Y A284307 Difference between pairs of elements:
%Y A284307   a(2n+1)-a(2n-1) = A010684(n-1), n > 0
%Y A284307 Compositions:
%Y A284307   a(n) = A133256(A116966(n-1)), n > 0
%Y A284307   a(A042948(n)) = A005408(n-1), n > 0
%Y A284307   A067060(a(n)) = A092486(n), n > 0
%K A284307 nonn,easy
%O A284307 1,2
%A A284307 _Guenther Schrack_, Mar 24 2017