A074067 -id:A074067 - cp's OEIS frontend

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

Reinhard Zumkeller, Aug 17 2002

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

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.

Cf. A002162, A064429, A074067, A074068.

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

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 *)

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

A074068 Zigzag modulo 7.

Original entry on oeis.org

1, 2, 3, 10, 9, 8, 7, 6, 5, 4, 17, 16, 15, 14, 13, 12, 11, 24, 23, 22, 21, 20, 19, 18, 31, 30, 29, 28, 27, 26, 25, 38, 37, 36, 35, 34, 33, 32, 45, 44, 43, 42, 41, 40, 39, 52, 51, 50, 49, 48, 47, 46, 59, 58, 57, 56, 55, 54, 53, 66, 65, 64, 63, 62, 61, 60, 73, 72, 71

Offset: 1

Reinhard Zumkeller, Aug 17 2002

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

Cf. A002162, A074066, A074067.

a074068 n = a074068_list !! (n-1)
a074068_list = 1 : 2 : 3 : xs where
   xs = 10 : 9 : 8 : 7 : 6 : 5 : 4 : map (+ 7) xs
-- Reinhard Zumkeller, Feb 21 2011

Range[3]~Join~Flatten[Reverse /@ Partition[Range[4, 73], 7]] (* after Harvey P. Dale at A074066, or *)
Range[3]~Join~Table[7 Floor[n/7] + 14 Floor[#/4] - # &@ Mod[n, 7], {n, 4, 69}] (* Michael De Vlieger, May 25 2016 *)
LinearRecurrence[{1,0,0,0,0,0,1,-1},{1,2,3,10,9,8,7,6,5,4,17},80] (* Harvey P. Dale, Jan 19 2025 *)

a(a(n)) = n (a self-inverse permutation).
For n > 1: a(n) = n iff n == 0 modulo 7.
a(n) = 7*floor(n/7) + 14*floor((n mod 7)/4) - (n mod 7) for n > 3; a(n) = n for n <= 3.
a(n) = a(n-1) + a(n-7) - a(n-8) for n > 11. - Chai Wah Wu, May 25 2016
G.f.: x+2*x^2+3*x^3 + x^4*(10-x-x^2-x^3-x^4-x^5-x^6+3*x^7) / ( (x^6+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

cp's OEIS Frontend

A074066 Zigzag modulo 3.

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