A263426 Permutation of the nonnegative integers: [4k+2, 4k+1, 4k, 4k+3, ...].

2, 1, 0, 3, 6, 5, 4, 7, 10, 9, 8, 11, 14, 13, 12, 15, 18, 17, 16, 19, 22, 21, 20, 23, 26, 25, 24, 27, 30, 29, 28, 31, 34, 33, 32, 35, 38, 37, 36, 39, 42, 41, 40, 43, 46, 45, 44, 47, 50, 49, 48, 51, 54, 53, 52, 55, 58, 57, 56, 59, 62, 61, 60, 63, 66, 65, 64

Offset: 0

Author

Wesley Ivan Hurt, Oct 17 2015

Keywords

• nonn
• easy

Comments

Fixed points are the odd numbers (A005408).

Links

• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).
• Index entries for sequences that are permutations of the natural numbers.

Crossrefs

Cf. A004442, A005408, A092486, A132429, A176742, A244009, A263449.

Programs

Magma

[n+(1+(-1)^n)*(-1)^(n*(n+1) div 2) : n in [0..80]];

Magma

/* By definition: */ &cat[[4*k+2,4*k+1,4*k,4*k+3]: k in [0..20]]; // Bruno Berselli, Nov 08 2015

Maple

A263426:=n->n + (1 + (-1)^n)*(-1)^(n*(n + 1)/2): seq(A263426(n), n=0..80);

Mathematica

Table[n + (1 + (-1)^n)*(-1)^(n*(n + 1)/2), {n, 0, 80}]

PARI

Vec((2-3*x+2*x^2+x^3)/((x-1)^2*(1+x^2)) + O(x^100)) \\ Altug Alkan, Oct 19 2015

Formula

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

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

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

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

a(n) = A092486(n) - 1.

a(n) = n + A176742(n) for n>0.

a(2n) = 2*A004442(n), a(2n+1) = A005408(n).

a(-n-1) = -A263449(n).

a(n+1) = a(n) - A132429(n+1)*(-1)^n.

Sum_{n>=0, n!=2} (-1)^(n+1)/a(n) = 1 - log(2) (A244009). - Amiram Eldar, Dec 25 2023