A065165 Permutation t->t+2 of Z, folded to N.

4, 6, 2, 8, 1, 10, 3, 12, 5, 14, 7, 16, 9, 18, 11, 20, 13, 22, 15, 24, 17, 26, 19, 28, 21, 30, 23, 32, 25, 34, 27, 36, 29, 38, 31, 40, 33, 42, 35, 44, 37, 46, 39, 48, 41, 50, 43, 52, 45, 54, 47, 56, 49, 58, 51, 60, 53, 62, 55, 64, 57, 66, 59, 68, 61, 70, 63, 72, 65, 74, 67, 76

Offset: 1

Author

Antti Karttunen, Oct 19 2001

Keywords

• nonn
• easy

Comments

Corresponds to simple periodic asynchronic site swap pattern ...222222... (holding a ball in each hand forever).

This permutation consists of just two infinite cycles.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Joe Buhler and R. L. Graham, Juggling Drops and Descents, Amer. Math. Monthly, 101, (no. 6) 1994, 507-519.
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
• Index entries for sequences that are permutations of the natural numbers.

Crossrefs

Row 2 of A065167. Inverse permutation: A065169.

Programs

Mathematica

CoefficientList[Series[(3 x^5 - 3 x^4 + 4 x^3 - 8 x^2 + 2 x + 4)/((x - 1)^2 (x + 1)), {x, 0, 100}], x] (* Vincenzo Librandi, Mar 08 2014 *)
LinearRecurrence[{1,1,-1},{4,6,2,8,1,10},80] (* Harvey P. Dale, May 09 2018 *)

PARI

Vec(x*(3*x^5-3*x^4+4*x^3-8*x^2+2*x+4)/((x-1)^2*(x+1))  + O(x^100)) \\ Colin Barker, Mar 07 2014

Formula

Let f: Z -> N be given by f(z) = 2z if z>0 else 2|z|+1, with inverse g(z) = z/2 if z even else (1-z)/2. Then a(n) = f(g(n)+2).

G.f.: x*(3*x^5-3*x^4+4*x^3-8*x^2+2*x+4) / ((x-1)^2*(x+1)). - Colin Barker, Feb 18 2013

a(n) = 4*(-1)^n+n for n>3. a(n) = a(n-1)+a(n-2)-a(n-3) for n>6. - Colin Barker, Mar 07 2014

Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) + 3/2. - Amiram Eldar, Aug 08 2023