A263313 Permutation of the nonnegative integers: [4k+3, 4k, 4k+1, 4k+2, ...].

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, 25, 26, 31, 28, 29, 30, 35, 32, 33, 34, 39, 36, 37, 38, 43, 40, 41, 42, 47, 44, 45, 46, 51, 48, 49, 50, 55, 52, 53, 54, 59, 56, 57, 58, 63, 60, 61, 62, 67, 64, 65

Offset: 0

Author

Wesley Ivan Hurt, Oct 19 2015

Keywords

• nonn
• easy

Links

• 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

Cf. A002162, A005843, A166519.

Programs

Magma

[n+1-2*(-1)^((n+1)*(n+2)*(n+3) div 2) : n in [0..100]];

Maple

A263313:=n->n+1-2*(-1)^((n+1)*(n+2)*(n+3)/2): seq(A263313(n), n=0..100);

Mathematica

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]
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 *)

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

Formula

G.f.: (3-3*x+x^2+x^3+2*x^4)/((x-1)^2*(1+x+x^2+x^3)).

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

a(n) = n+1-2*(-1)^((n+1)*(n+2)*(n+3)/2).

a(2n) = A166519(n), a(2n+1) = A005843(n).

a(n) = n+(-1)^n+2*cos(n*Pi/2). - Wesley Ivan Hurt, May 09 2021

Sum_{n>=0, n!=1} (-1)^n/a(n) = log(2) (A002162). - Amiram Eldar, Dec 25 2023