A064429 a(n) = floor(n / 3) * 3 + sign(n mod 3) * (3 - n mod 3).

0, 2, 1, 3, 5, 4, 6, 8, 7, 9, 11, 10, 12, 14, 13, 15, 17, 16, 18, 20, 19, 21, 23, 22, 24, 26, 25, 27, 29, 28, 30, 32, 31, 33, 35, 34, 36, 38, 37, 39, 41, 40, 42, 44, 43, 45, 47, 46, 48, 50, 49, 51, 53, 52, 54, 56, 55, 57, 59, 58, 60, 62, 61, 63, 65, 64, 66, 68, 67, 69, 71, 70

Offset: 0

Author

Reinhard Zumkeller, Oct 15 2001

Keywords

• nonn
• easy

Comments

a(a(n)) = n (a self-inverse permutation).

Take natural numbers, exchange trisections starting with 1 and 2.

Lodumo_3 of A080425. - Philippe Deléham, Apr 26 2009

From Franck Maminirina Ramaharo, Jul 27 2018: (Start)

The sequence is A008585 interleaved with A016789 and A016777.

a(n) is also obtained as follows: write n in base 3; if the rightmost digit is '1', then replace it with '2' and vice versa; convert back to decimal. For example a(14) = a('11'2') = '11'1' = 13 and a(10) = a('10'1') = '10'2' = 11. (End)

A permutation of the nonnegative integers partitioned into triples [3*k-3, 3*k-1, 3*k-2] for k > 0. - Guenther Schrack, Feb 05 2020

Examples

			From _Franck Maminirina Ramaharo_, Jul 27 2018: (Start)
Interleave 3 sequences:
A008585: 0.....3.....6.....9.......12.......15........
A016789: ..2.....5.....8.....11.......14.......17.....
A016777: ....1.....4.....7......10.......13.......16..
(End)
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..3000
• Eric Weisstein's World of Mathematics, Alternating Permutations.
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
• Index entries for sequences that are permutations of the natural numbers.

Crossrefs

Cf. A001477, A004442, A074066, A080425, A080782, A102283, A143097.

Programs

GAP

a:=[0,2,1,3];; for n in [5..100] do a[n]:=a[n-1]+a[n-3]-a[n-4]; od; a; # Muniru A Asiru, Jul 27 2018

Magma

[2*n - 3 - 3*((n-2) div 3): n in [0..80]]; // Vincenzo Librandi, Aug 05 2018

Maple

A064429:=n->2*n-3-3*floor((n-2)/3): seq(A064429(n), n=0..100); # Wesley Ivan Hurt, Nov 30 2013

Mathematica

Table[2 n - 3 - 3 Floor[(n - 2)/3], {n, 0, 100}] (* Wesley Ivan Hurt, Nov 30 2013 *)
{#+1,#-1,#}[[Mod[#,3,1]]]&/@Range[0, 100] (* Federico Provvedi, May 11 2021 *)
LinearRecurrence[{1,0,1,-1},{0,2,1,3},80] (* or *) {#[[1]],#[[3]],#[[2]]}&/@Partition[Range[0,80],3]//Flatten (* Harvey P. Dale, Mar 28 2025 *)

PARI

a(n) = 2*n-3-3*((n-2)\3); \\ Altug Alkan, Oct 06 2017

Formula

a(n) = A080782(n+1) - 1.

a(n) = n - 2*sin(4*Pi*n/3)/sqrt(3). - Jaume Oliver Lafont, Dec 05 2008

a(n) = A001477(n) + A102283(n). - Jaume Oliver Lafont, Dec 05 2008

a(n) = lod_3(A080425(n)). - Philippe Deléham, Apr 26 2009

G.f.: x*(2 - x + 2*x^2)/((1 + x + x^2)*(1 - x)^2 ). - R. J. Mathar, Feb 20 2011

a(n) = 2*n - 3 - 3*floor((n-2)/3). - Wesley Ivan Hurt, Nov 30 2013

a(n) = a(n-1) + a(n-3) - a(n-4) for n > 3. - Wesley Ivan Hurt, Oct 06 2017

E.g.f.: x*exp(x) + (2*sin((sqrt(3)*x)/2))/(exp(x/2)*sqrt(3)). - Franck Maminirina Ramaharo, Jul 27 2018

From Guenther Schrack, Feb 05 2020: (Start)

a(n) = a(n-3) + 3 with a(0)=0, a(1)=2, a(2)=1 for n > 2;

a(n) = n + (w^(2*n) - w^n)*(1 + 2*w)/3 where w = (-1 + sqrt(-3))/2. (End)

Sum_{n>=1} (-1)^n/a(n) = log(2)/3. - Amiram Eldar, Jan 31 2023