A074066 Zigzag modulo 3.

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

Offset: 1

Author

Reinhard Zumkeller, Aug 17 2002

Keywords

• nonn
• easy
• nice

Comments

Take natural numbers, exchange trisections starting with 2 and 4.

Links

• Eric Weisstein's World of Mathematics, Alternating Permutations.
• Reinhard Zumkeller, Illustration for A074066-A074068.
• 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. A002162, A064429, A074067, A074068.

Programs

Haskell

a074066 n = a074066_list !! (n-1)
a074066_list = 1 : xs where xs = 4 : 3 : 2 : map (+ 3) xs
-- Reinhard Zumkeller, Feb 21 2011

Mathematica

a[n_] := n + Mod[n, 3]*(3*Mod[n, 3] - 5); a[1] = 1; Table[a[n], {n, 1, 69}] (* Jean-François Alcover, Nov 04 2011 *)
Join[{1},Flatten[Reverse/@Partition[Range[2,73],3]]] (* Harvey P. Dale, Feb 17 2012 *)

Formula

a(1)=1; for n>0: a(3*n-1) = 3*n+1, a(3*n) = 3*n, a(3*n+1) = 3*n-1.

a(a(n))=n (self-inverse permutation); for n>1: a(n) = n iff n == 0 modulo 3.

For n > 1: a(n) = 3*floor(n/3) + (n mod 3)^2 * (-1)^(n mod 3); a(1)=1.

a(n) = a(n-1) + a(n-3) - a(n-4) for n > 5. - Chai Wah Wu, May 25 2016

For n > 1, a(n) = n - (4/sqrt(3))*sin(2*n*Pi/3). - Wesley Ivan Hurt, Sep 29 2017

g.f.: x + x^2*(4-x-x^2+x^3) / ( (1+x+x^2)*(x-1)^2 ). - R. J. Mathar, May 22 2019

Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) (A002162). - Amiram Eldar, Dec 24 2023