A298364 Permutation of the natural numbers partitioned into quadruples [4k-2, 4k-1, 4k-3, 4k] for k > 0.

2, 3, 1, 4, 6, 7, 5, 8, 10, 11, 9, 12, 14, 15, 13, 16, 18, 19, 17, 20, 22, 23, 21, 24, 26, 27, 25, 28, 30, 31, 29, 32, 34, 35, 33, 36, 38, 39, 37, 40, 42, 43, 41, 44, 46, 47, 45, 48, 50, 51, 49, 52, 54, 55, 53, 56, 58, 59, 57, 60, 62, 63, 61, 64, 66, 67, 65

Offset: 1

Author

Guenther Schrack, Jan 18 2018

Keywords

• nonn
• easy

Comments

Partition the natural number sequence into quadruples starting with (1,2,3,4); swap the first and second elements, then swap the second and third elements; repeat for all quadruples.

Links

• Guenther Schrack, Table of n, a(n) for n = 1..10000
• Index entries for sequences that are permutations of the natural numbers
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Inverse: A292576.

Sequence of fixed points: A008586(n) for n > 0.

First differences: (-1)^floor(n^2/4)*A068073(n-1) for n > 0.

Subsequences:

elements with odd index: A042963(A103889(n)) for n > 0.

elements with even index A014601(n) for n > 0.

odd elements: A166519(n-1) for n > 0.

indices of odd elements: A042964(n) for n > 0.

even elements: A005843(n) for n > 0.

indices of even elements: A042948(n) for n > 0.

Other similar permutations: A116966, A284307, A292576.

Programs

MATLAB

a = [2 3 1 4];
max = 10000;    % Generation of b-file.
for n := 5:max
   a(n) = a(n-4) + 4;
end;

Mathematica

Nest[Append[#, #[[-4]] + 4] &, {2, 3, 1, 4}, 63] (* or *)
Array[# + ((-1)^# + ((-1)^(# (# - 1)/2)) (1 - 2 (-1)^#))/2 &, 67] (* Michael De Vlieger, Jan 23 2018 *)
LinearRecurrence[{1,0,0,1,-1},{2,3,1,4,6},70] (* Harvey P. Dale, Dec 12 2018 *)

PARI

for(n=1, 100, print1(n + ((-1)^n + ((-1)^(n*(n-1)/2))*(1 - 2*(-1)^n))/2, ", "))

Formula

O.g.f.: (3*x^3 - 2*x^2 + x + 2)/(x^5 - x^4 - x - 1).

a(1) = 2, a(2) = 3, a(3) = 1, a(4) = 4, a(n) = a(n-4) + 4 for n > 4.

a(n) = n + ((-1)^n + ((-1)^(n*(n-1)/2))*(1 - 2*(-1)^n))/2.

a(n) = n + (cos(n*Pi) - cos(n*Pi/2) + 3*sin(n*Pi/2))/2.

a(n) = 2*floor((n+1)/2) - 4*floor((n+1)/4) + floor(n/2) + 2*floor(n/4).

a(n) = n + (-1)^floor((n-1)^2/4)*A140081(n) for n > 0.

a(n) = A056699(n+1) - 1, n > 0.

a(n+2) = A168269(n+1) - a(n), n > 0.

a(n+2) = a(n) + (-1)^floor((n+1)^2/4)*A132400(n+2) for n > 0.

Linear recurrence: a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.

First differences: periodic, (1, -2, 3, 2) repeat.

Compositions:

a(n) = A080412(A116966(n-1)) for n > 0.

a(n) = A284307(A256008(n)) for n > 0.

a(A067060(n)) = A133256(n) for n > 0.

A116966(a(n+1)-1) = A092486(n) for n >= 0.

A056699(a(n)) = A256008(n) for n > 0.