A039720 Period of n-countdown club-passing juggling pattern.

6, 8, 30, 24, 70, 48, 126, 80, 198, 120, 286, 168, 390, 224, 510, 288, 646, 360, 798, 440, 966, 528, 1150, 624, 1350, 728, 1566, 840, 1798, 960, 2046, 1088, 2310, 1224, 2590, 1368, 2886, 1520, 3198, 1680, 3526, 1848, 3870, 2024, 4230, 2208, 4606, 2400

Offset

2,1

Comments

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.

Links

Harvey P. Dale, Table of n, a(n) for n = 2..1000

Martin Frost, An article in rec.juggling

Rob Street, An article in rec.juggling

Tarim, An article in rec.juggling

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Formula

a(n) = n^2-1 for n odd, = 2(n^2-1) for n even.

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

a(n) = LCM(2n+2, 2n-2). - Wesley Ivan Hurt, Jan 05 2014

a(n) = (n^2 - 1) * 2^(n+1 mod 2). - Wesley Ivan Hurt, Jan 05 2014

a(n) = (n+1) * (n-1) * (1 + ((n+1) mod 2)). - Wesley Ivan Hurt, Jan 05 2014

a(n) = A005563(n-1) * A000034(n-1). - Wesley Ivan Hurt, Jan 05 2014

Example

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.

Maple

A039720:=n->(3+(-1)^n)*(n^2-1)/2; seq(A039720(n), n=2..100); # Wesley Ivan Hurt, Jan 05 2014

Mathematica

Table[c=n^2-1; If[OddQ[n], c, 2c], {n, 2, 50}] (* Harvey P. Dale, Nov 27 2012 *)

Prog

(PARI) a(n)=if(n%2, 1, 2)*(n^2-1) \\ Charles R Greathouse IV, Feb 10 2017

Crossrefs

Sequence in context: A345003 A000773 A258283 * A056097 A099431 A323201

Adjacent sequences: A039717 A039718 A039719 * A039721 A039722 A039723

Keyword

easy,nonn,nice

Author

Mark Tillotson (markt(AT)chaos.org.uk)

Status

approved