A004442 Natural numbers, pairs reversed: a(n) = n + (-1)^n; also Nimsum n + 1.

1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14, 17, 16, 19, 18, 21, 20, 23, 22, 25, 24, 27, 26, 29, 28, 31, 30, 33, 32, 35, 34, 37, 36, 39, 38, 41, 40, 43, 42, 45, 44, 47, 46, 49, 48, 51, 50, 53, 52, 55, 54, 57, 56, 59, 58, 61, 60, 63, 62, 65, 64, 67, 66, 69

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

A self-inverse permutation of the natural numbers.

Nonnegative numbers rearranged with least disturbance to maintain a(n) not equal to n. - Amarnath Murthy, Sep 13 2002

Essentially lodumo_2 of A059841. - Philippe Deléham, Apr 26 2009

a(n) = A180176(n) for n >= 20. - Reinhard Zumkeller, Aug 15 2010

References

• E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 60.
• J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
• Index entries for sequences related to Nim-sums
• Index entries for sequences that are permutations of the natural numbers
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A003987, A004443, A004444. Equals A014681 - 1.

Cf. A005843, A005408, A059841.

Cf. A002162

Programs

Haskell

import Data.List (transpose)
import Data.Bits (xor)
a004442 = xor 1 :: Integer -> Integer
a004442_list = concat $ transpose [a005408_list, a005843_list]
-- Reinhard Zumkeller, Jun 23 2013, Feb 01 2013, Oct 20 2011

Maple

a[0]:=1:a[1]:=0:for n from 2 to 70 do a[n]:=a[n-2]+2 od: seq(a[n], n=0..68); # Zerinvary Lajos, Feb 19 2008

Mathematica

Table[n + (-1)^n, {n, 0, 72}] (* or *)
CoefficientList[Series[(1 - x + 2x^2)/((1 - x)(1 - x^2)), {x, 0, 72}], x] (* Robert G. Wilson v, Jun 16 2006 *)
Flatten[Reverse/@Partition[Range[0,69],2]] (* or *) LinearRecurrence[{1,1,-1},{1,0,3},70] (* Harvey P. Dale, Jul 29 2018 *)

PARI

a(n)=n+(-1)^n \\ Charles R Greathouse IV, Nov 20 2012

PARI

Vec((1-x+2*x^2)/((1-x)*(1-x^2)) + O(x^100)) \\ Altug Alkan, Feb 04 2016

Python

def a(n): return n^1
print([a(n) for n in range(69)]) # Michael S. Branicky, Jan 23 2022

Formula

a(n) = n XOR 1. - Odimar Fabeny, Sep 05 2004

G.f.: (1-x+2x^2)/((1-x)*(1-x^2)). - Mitchell Harris, Jan 10 2005

a(n+1) = lod_2(A059841(n)). - Philippe Deléham, Apr 26 2009

a(n) = 2*n - a(n-1) - 1 with n > 0, a(0)=1. - Vincenzo Librandi, Nov 18 2010

a(n) = Sum_{k=1..n-1} (-1)^(n-1-k)*C(n+1,k). - Mircea Merca, Feb 07 2013

For n > 1, a(n)^a(n) == 1 (mod n). - Thomas Ordowski, Jan 04 2016

Sum_{n>=0,n<>1} (-1)^n/a(n) = log(2) = A002162. - Peter McNair, Aug 07 2023

Extensions

Offset adjusted by Reinhard Zumkeller, Mar 05 2010