OFFSET
0,3
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 linear recurrences with constant coefficients, signature (1,1,-1).
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
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
Offset adjusted by Reinhard Zumkeller, Mar 05 2010
STATUS
approved