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A065179 Number of site swap patterns with 3 balls and exact period n.

%I A065179 #18 Apr 14 2025 07:39:51
%S A065179 1,3,12,42,156,554,2028,7350,26936,98874,365196,1353520,5039580,
%T A065179 18831306,70626140,265741350,1002984060,3796211692,14406086604,
%U A065179 54801192684,208932673508,798218225802,3055417434732,11716355452900
%N A065179 Number of site swap patterns with 3 balls and exact period n.
%H A065179 G. C. Greubel, <a href="/A065179/b065179.txt">Table of n, a(n) for n = 1..1000</a>
%H A065179 Juggling Information Service, <a href="http://www.juggling.org/bin/mfs/JIS/help/siteswap/">Site Swap FAQs</a>
%H A065179 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.
%F A065179 G.f.: Sum_{k>=1} mu(k) * log(1 + x^k/(1 - 4*x^k))/k. - _Seiichi Manyama_, Apr 14 2025
%e A065179 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 A065179 [seq(DistSS(p,3),p=1..60)];
%p A065179 A065179 := proc(n)
%p A065179     add( mobius(n/d)*(4^d-3^d),d=numtheory[divisors](n)) /n ;
%p A065179 end proc:
%p A065179 seq(A065179(n),n=1..30) ; # _R. J. Mathar_, Aug 05 2015
%t A065179 a[n_] := DivisorSum[n, MoebiusMu[n/#]*(4^# - 3^#)&]/n; Array[a, 25] (* _Jean-François Alcover_, Mar 06 2016 *)
%Y A065179 Row 3 of A065177.
%Y A065179 Cf. A008683, A065178, A065180.
%K A065179 nonn
%O A065179 1,2
%A A065179 _Antti Karttunen_, Oct 19 2001