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

1,1

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

Table of n, a(n) for n=1..72.

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).

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

Mathematica

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

Prog

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

Crossrefs

Essentially the same as A014681.

Odd numbers: A005408. Even numbers: A005843.

Cf. A004442.

Sequence in context: A167419 A114285 A014681 * A137805 A163501 A306229

Adjacent sequences: A103886 A103887 A103888 * A103890 A103891 A103892

Keyword

nonn,easy

Author

Zak Seidov, Feb 20 2005

Status

approved