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Number of site swap patterns with 3 balls and exact period n.
4

%I #13 Dec 11 2017 02:39:23

%S 1,3,12,42,156,554,2028,7350,26936,98874,365196,1353520,5039580,

%T 18831306,70626140,265741350,1002984060,3796211692,14406086604,

%U 54801192684,208932673508,798218225802,3055417434732,11716355452900

%N Number of site swap patterns with 3 balls and exact period n.

%H G. C. Greubel, <a href="/A065179/b065179.txt">Table of n, a(n) for n = 1..1000</a>

%H Juggling Information Service, <a href="http://www.juggling.org/bin/mfs/JIS/help/siteswap/">Site Swap FAQs</a>

%H M. Macauley, <a href="http://www.math.hmc.edu/seniorthesis/archives/2003/mmacaule/mmacaule-2003-thesis.pdf">Braids and Juggling patterns</a>, Thesis (Harvey Mudd College) 2003, equation A1.

%e We have one period 1 (3, the three-ball cascade), three period two (42/24, 51/15 = three-ball shower and 60/06) and twelve period three 3-ball siteswaps (423, 441, 450, 522, 531, 603, 612, 630, 711, 720, 801, 900) (The average of digits is always 3).

%p [seq(DistSS(p,3),p=1..60)];

%p A065179 := proc(n)

%p add( mobius(n/d)*(4^d-3^d),d=numtheory[divisors](n)) /n ;

%p end proc:

%p seq(A065179(n),n=1..30) ; # _R. J. Mathar_, Aug 05 2015

%t a[n_] := DivisorSum[n, MoebiusMu[n/#]*(4^# - 3^#)&]/n; Array[a, 25] (* _Jean-François Alcover_, Mar 06 2016 *)

%Y Row 3 of A065177. Cf. A065178, A065180.

%K nonn

%O 1,2

%A _Antti Karttunen_, Oct 19 2001