A103889 Odd and even positive integers swapped.

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, 69, 72, 71

Offset: 1

Author

Zak Seidov, Feb 20 2005

Keywords

• nonn
• easy

Comments

For n >= 5, also the number of (undirected) Hamiltonian cycles in the (n-2)-Moebius ladder. - Eric W. Weisstein, May 06 2019

For n >= 4, also the number of (undirected) Hamiltonian cycles in the (n-1)-prism graph. - Eric W. Weisstein, May 06 2019

The lexicographically first involution of the natural numbers with no fixed points. - Alexander Fraebel, Sep 08 2020

Links

• Eric Gottlieb, Matjaž Krnc, and Peter Muršič, Nim on Integer Partitions and Hyperrectangles, arXiv:2506.04991 [math.CO], 2025. See p. 16.
• Eric Weisstein's World of Mathematics, Hamiltonian Cycle
• Eric Weisstein's World of Mathematics, Moebius Ladder
• Eric Weisstein's World of Mathematics, Prism Graph
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Essentially the same as A014681.

Odd numbers: A005408. Even numbers: A005843.

Cf. A004442.

Programs

Haskell

import Data.List (transpose)
a103889 n = n - 1 + 2 * mod n 2
a103889_list = concat $ transpose [tail a005843_list, a005408_list]
-- Reinhard Zumkeller, Jun 23 2013, Feb 21 2011

Magma

[n eq 1 select 2 else -Self(n-1)+2*n-1: n in [1..72]];

Mathematica

Table[{n + 1, n}, {n, 1, 100, 2}] // Flatten
Table[n - (-1)^n, {n, 25}] (* Eric W. Weisstein, May 06 2019 *)

PARI

a(n)=n-1+if(n%2,2) \\ Charles R Greathouse IV, Feb 24 2011

Python

def a(n): return n+1 if n&1 else n-1
print([a(n) for n in range(1, 73)]) # Michael S. Branicky, May 03 2023

Formula

a(2k) = 2k-1 = A005408(k), a(2k-1) = 2k = A005843(k), k=1, 2, ...

O.g.f.: x*(x^2-x+2)/((x-1)^2*(1+x)). - R. J. Mathar, Apr 06 2008

a(n) = n-1+2*(n mod 2). - Rolf Pleisch, Apr 22 2008

a(n) = 2*n-a(n-1)-1 (with a(1)=2). - Vincenzo Librandi, Nov 16 2010

From Bruno Berselli, Nov 16 2010: (Start)

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

a(n) - a(n-1) - a(n-2) + a(n-3) = 0 for n > 3.

(a(n) - 1)*(a(n-1) + 1) = 2*A176222(n+1) for n > 1.

(a(n) - 1)*(a(n-3) + 1) = 2*A176222(n) for n > 3. (End)

E.g.f.: 1 - exp(-x) + x*exp(x). - Stefano Spezia, May 03 2023