A065170 Permutation t->t-3 of Z, folded to N.

7, 5, 9, 3, 11, 1, 13, 2, 15, 4, 17, 6, 19, 8, 21, 10, 23, 12, 25, 14, 27, 16, 29, 18, 31, 20, 33, 22, 35, 24, 37, 26, 39, 28, 41, 30, 43, 32, 45, 34, 47, 36, 49, 38, 51, 40, 53, 42, 55, 44, 57, 46, 59, 48, 61, 50, 63, 52, 65, 54, 67, 56, 69, 58, 71, 60, 73, 62, 75, 64, 77, 66

Offset: 1

Author

Antti Karttunen, Oct 19 2001

Keywords

• nonn
• easy

Comments

This permutation consists of just three cycles, which are infinite.

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

Inverse permutation to A065166.

Programs

Mathematica

CoefficientList[Series[(x^8 - x^7 + 4 x^6 - 4 x^5 + 4 x^4 - 4 x^3 - 3 x^2 - 2 x + 7)/((x - 1)^2 (x + 1)), {x, 0, 100}], x] (* Vincenzo Librandi, Mar 08 2014 *)
LinearRecurrence[{1,1,-1},{7,5,9,3,11,1,13,2,15},80] (* Harvey P. Dale, Oct 19 2018 *)

PARI

Vec(x*(x^8-x^7+4*x^6-4*x^5+4*x^4-4*x^3-3*x^2-2*x+7)/((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)-3).

G.f.: x*(x^8-x^7+4*x^6-4*x^5+4*x^4-4*x^3-3*x^2-2*x+7) / ((x-1)^2*(x+1)). - Colin Barker, Feb 18 2013

a(n) = -6*(-1)^n+n for n>6. a(n) = a(n-1)+a(n-2)-a(n-3) for n>9. - Colin Barker, Mar 07 2014

Sum_{n>=1} (-1)^n/a(n) = 46/15 - log(2). - Amiram Eldar, Aug 08 2023