A014681 Fix 0; exchange even and odd numbers.

0, 2, 1, 4, 3, 6, 5, 8, 7, 10, 9, 12, 11, 14, 13, 16, 15, 18, 17, 20, 19, 22, 21, 24, 23, 26, 25, 28, 27, 30, 29, 32, 31, 34, 33, 36, 35, 38, 37, 40, 39, 42, 41, 44, 43, 46, 45, 48, 47, 50, 49, 52, 51, 54, 53, 56, 55, 58, 57, 60, 59, 62, 61, 64, 63, 66, 65, 68, 67, 70

Offset: 0

Author

Mohammad K. Azarian

Keywords

• nonn
• easy

Comments

A self-inverse permutation of the nonnegative numbers.

If we ignore the first term 0, then this can be obtained as: a(n) is the smallest number different from n, not occurring earlier and coprime to n. - Amarnath Murthy, Apr 16 2003 [Corrected by Alois P. Heinz, May 06 2015]

a(0)=0, a(1)=2, then repeatedly subtract 1 and then add 3. - Jon Perry, Aug 12 2014

The biggest term of the pair [a(n), a(n+1)] is always even. This is the lexicographically first sequence with this property starting with a(1) = 0 and always extented with the smallest integer not yet present. - Eric Angelini, Feb 20 2017

Links

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

Crossrefs

Composing this permutation with A065190 gives A065164.

Equals 1 + A004442.

Cf. A103889.

Programs

Mathematica

Table[n - (-1)^n, {n, 1, 60}]
Join[{0},LinearRecurrence[{1, 1, -1},{2, 1, 4},69]] (* Ray Chandler, Sep 03 2015 *)

PARI

a(n)=n - (-1)^n \\ Charles R Greathouse IV, May 06 2015

Formula

G.f.: x*(2-x+x^2)/((1-x)*(1-x^2)). - N. J. A. Sloane

a(n) = n - (-1)^n = a(n-1) + a(n-2) - a(n-3) = a(n-2) + 2. - Henry Bottomley, Mar 29 2000

a(0) = 0; a(2m+1) = 2m+2; for m > 0 a(2m) = 2m - 1. - George E. Antoniou, Dec 04 2001

a(n) = n - (-1)^n + 0^n for n >= 0. - Bruno Berselli, Nov 16 2010

E.g.f.: 1 + (x - 1)*cosh(x) + (1 + x)*sinh(x). - Stefano Spezia, Sep 02 2022