%I #19 Feb 10 2017 00:30:30
%S 6,8,30,24,70,48,126,80,198,120,286,168,390,224,510,288,646,360,798,
%T 440,966,528,1150,624,1350,728,1566,840,1798,960,2046,1088,2310,1224,
%U 2590,1368,2886,1520,3198,1680,3526,1848,3870,2024,4230,2208,4606,2400
%N Period of n-countdown club-passing juggling pattern.
%C Tarim's countdown: 2 people pass clubs to each other after n throws, then n-1 throws, then n-2, ..., 2, 1, 2, ... n-1. Thus for 3-countdown it is 3,2,1,2 (and repeat), or put in terms of pass throws and self-throws, pass-self-self-pass-self-pass-pass-self and repeat.
%H Harvey P. Dale, <a href="/A039720/b039720.txt">Table of n, a(n) for n = 2..1000</a>
%H Martin Frost, <a href="http://www.juggling.org/rj/1998/09/08-080230">An article in rec.juggling</a>
%H Rob Street, <a href="http://www.juggling.org/bin/mfs/JIS/rj/1996/04/17-232435">An article in rec.juggling</a>
%H Tarim, <a href="http://www.juggling.org/bin/mfs/JIS/rj/1998/09/06-225057">An article in rec.juggling</a>
%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).
%F a(n) = n^2-1 for n odd, = 2(n^2-1) for n even.
%F a(n) = (3+(-1)^n)*(-1+n^2)/2. a(n) = 3*a(n-2)-3*a(n-4)+a(n-6). G.f.: 2*x^2*(x^4-6*x^2-4*x-3) / ((x-1)^3*(x+1)^3). - _Colin Barker_, Feb 18 2013
%F a(n) = LCM(2n+2, 2n-2). - _Wesley Ivan Hurt_, Jan 05 2014
%F a(n) = (n^2 - 1) * 2^(n+1 mod 2). - _Wesley Ivan Hurt_, Jan 05 2014
%F a(n) = (n+1) * (n-1) * (1 + ((n+1) mod 2)). - _Wesley Ivan Hurt_, Jan 05 2014
%F a(n) = A005563(n-1) * A000034(n-1). - _Wesley Ivan Hurt_, Jan 05 2014
%e a(6) = 70 because 6^2 - 1 = 35 is odd, so it is doubled to 70 because the passing pattern must begin and end with same hand.
%p A039720:=n->(3+(-1)^n)*(n^2-1)/2; seq(A039720(n), n=2..100); # _Wesley Ivan Hurt_, Jan 05 2014
%t Table[c=n^2-1;If[OddQ[n],c,2c],{n,2,50}] (* _Harvey P. Dale_, Nov 27 2012 *)
%o (PARI) a(n)=if(n%2, 1, 2)*(n^2-1) \\ _Charles R Greathouse IV_, Feb 10 2017
%K easy,nonn,nice
%O 2,1
%A Mark Tillotson (markt(AT)chaos.org.uk)