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

1,1

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.

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

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 *)

Prog

(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

Crossrefs

Row 2 of A065167. Inverse permutation: A065169.

Sequence in context: A336049 A153124 A066252 * A088516 A114538 A181096

Adjacent sequences: A065162 A065163 A065164 * A065166 A065167 A065168

Keyword

nonn,easy

Author

Antti Karttunen, Oct 19 2001

Status

approved