A065165 - OEIS

%I #28 Aug 08 2023 03:21:41

%S 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,

%T 23,32,25,34,27,36,29,38,31,40,33,42,35,44,37,46,39,48,41,50,43,52,45,

%U 54,47,56,49,58,51,60,53,62,55,64,57,66,59,68,61,70,63,72,65,74,67,76

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

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

%C This permutation consists of just two infinite cycles.

%H Vincenzo Librandi, <a href="/A065165/b065165.txt">Table of n, a(n) for n = 1..1000</a>

%H Joe Buhler and R. L. Graham, <a href="http://www.cecm.sfu.ca/organics/papers/buhler/index.html">Juggling Drops and Descents</a>, Amer. Math. Monthly, 101, (no. 6) 1994, 507-519.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.

%F 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).

%F 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

%F 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

%F Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) + 3/2. - _Amiram Eldar_, Aug 08 2023

%t 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 *)

%t LinearRecurrence[{1,1,-1},{4,6,2,8,1,10},80] (* _Harvey P. Dale_, May 09 2018 *)

%o (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

%Y Row 2 of A065167. Inverse permutation: A065169.

%K nonn,easy

%O 1,1

%A _Antti Karttunen_, Oct 19 2001