A263449 Permutation of the natural numbers: [4k+1, 4k+4, 4k+3, 4k+2, ...].

1, 4, 3, 2, 5, 8, 7, 6, 9, 12, 11, 10, 13, 16, 15, 14, 17, 20, 19, 18, 21, 24, 23, 22, 25, 28, 27, 26, 29, 32, 31, 30, 33, 36, 35, 34, 37, 40, 39, 38, 41, 44, 43, 42, 45, 48, 47, 46, 49, 52, 51, 50, 53, 56, 55, 54, 57, 60, 59, 58, 61, 64, 63, 62, 65, 68, 67

Offset: 0

Author

Wesley Ivan Hurt, Oct 18 2015

Keywords

• nonn
• easy

Comments

From Franklin T. Adams-Watters, Jul 13 2017: (Start)

For this to be a permutation, it should have offset one, not zero.

With offset 1, a(n) is the smallest positive integer == n (mod 2) with a(n) != a(n-1) + 1. (End)

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• 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. A002162, A005408, A014681.

Programs

Magma

[n+1+(1-(-1)^n)*(-1)^(n*(n-1) div 4) : n in [0..100]];

Magma

/* By definition: */ &cat[[4*k+1,4*k+4,4*k+3,4*k+2]: k in [0..20]]; // Bruno Berselli, Oct 19 2015

Maple

A263449:=n->n+1+(1-(-1)^n)*(-1)^(n*(n-1)/2): seq(A263449(n), n=0..100);

Mathematica

Table[n + 1 + (1 - (-1)^n) (-1)^(n (n - 1)/2), {n, 0, 100}] (* or *) LinearRecurrence[{2,-2,2,-1}, {1, 4, 3, 2}, 70]

PARI

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

PARI

a(n) = n+1+I*((-I)^n-I^n) \\ Colin Barker, Oct 27 2015

Formula

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

a(2n) = A005408(n), a(2n+1) = 2*A014681(n+1).

a(n) = n+1+i*((-i)^n-i^n), where i=sqrt(-1). - Colin Barker, Oct 27 2015

a(n) = 4*ceiling(n/4) - (n mod 4) + 1. - Wesley Ivan Hurt, Nov 07 2015

Sum_{n>=0} (-1)^n/a(n) = log(2) (A002162). - Amiram Eldar, Nov 28 2023