A074067 Zigzag modulo 5.

1, 2, 7, 6, 5, 4, 3, 12, 11, 10, 9, 8, 17, 16, 15, 14, 13, 22, 21, 20, 19, 18, 27, 26, 25, 24, 23, 32, 31, 30, 29, 28, 37, 36, 35, 34, 33, 42, 41, 40, 39, 38, 47, 46, 45, 44, 43, 52, 51, 50, 49, 48, 57, 56, 55, 54, 53, 62, 61, 60, 59, 58, 67, 66, 65, 64, 63, 72, 71

Offset: 1

Author

Reinhard Zumkeller, Aug 17 2002

Keywords

• nonn
• easy

Links

• Eric Weisstein's World of Mathematics, Alternating Permutations.
• Reinhard Zumkeller, Illustration for A074066-A074068.
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).
• Index entries for sequences that are permutations of the natural numbers.

Crossrefs

Cf. A002162, A074066, A074068.

Programs

Haskell

a074067 n = a074067_list !! (n-1)
a074067_list = 1 : 2 : xs where xs = 7 : 6 : 5 : 4 : 3 : map (+ 5) xs
-- Reinhard Zumkeller, Feb 21 2011

Mathematica

{1, 2}~Join~Flatten[Reverse /@ Partition[Range[3, 72], 5]] (* after Harvey P. Dale at A074066, or *)
{1, 2}~Join~Table[5 Floor[n/5] + 10 Floor[#/3] - # &@ Mod[n, 5], {n, 3, 69}] (* Michael De Vlieger, May 25 2016 *)
LinearRecurrence[{1,0,0,0,1,-1},{1,2,7,6,5,4,3,12},70] (* Harvey P. Dale, Jun 18 2025 *)

Formula

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

For n > 1: a(n) = n iff n == 0 modulo 5.

a(n) = 5*floor(n/5) + 10*floor((n mod 5)/3) - (n mod 5) for n > 2; a(n) = n for n <= 2.

a(n) = a(n-1) + a(n-5) - a(n-6) for n > 8. - Chai Wah Wu, May 25 2016

G.f.: x+2*x + x^3*(7-x-x^2-x^3-x^4+2*x^5) / ( (x^4+x^3+x^2+x+1)*(x-1)^2 ). - R. J. Mathar, May 22 2019

Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) (A002162). - Amiram Eldar, May 11 2025