A292576 Permutation of the natural numbers partitioned into quadruples [4k-1, 4k-3, 4k-2, 4k], k > 0.

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

Offset: 1

Author

Guenther Schrack, Sep 19 2017

Keywords

• nonn

Comments

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

Links

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

Crossrefs

Inverse: A056699(n+1) - 1 for n > 0.

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

Subsequences:

elements with odd index: A042964(A103889(n)) for n > 0.

elements with even index: A042948(n) for n > 0.

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

indices of odd elements: A042963(n) for n > 0.

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

indices of even elements: A014601(n) for n > 0.

Sum of pairs of elements:

a(n+2) + a(n) = A163980(n+1) = A168277(n+2) for n > 0.

Difference between pairs of elements:

a(n+2) - a(n) = (-1)^A011765(n+3)*A091084(n+1) for n > 0.

Compound relations:

a(n) = A284307(n+1) - 1 for n > 0.

a(n+2) - 2*a(n+1) + a(n) = (-1)^A011765(n)*A132400(n+1) for n > 0.

Compositions:

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

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

a(A042963(n)) = A166519(n-1) for n > 0.

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

Programs

MATLAB

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

PARI

for(n=1, 10000, print1(n + ((-1)^(n*(n-1)/2)*(2 - (-1)^n) - (-1)^n)/2, ", "))

Formula

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

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

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

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

a(n) = n + n mod 2 + (ceiling(n/2)) mod 2 - 2*(floor(n/2) mod 2).

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

First Differences, periodic: (-2, 1, 2, 3), repeat; also (-1)^A130569(n)*A068073(n+2) for n > 0.