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A065178 Number of site swap patterns with 2 balls and exact period n.

%I A065178 #29 Apr 14 2025 07:39:45
%S A065178 1,2,6,15,42,107,294,780,2128,5781,15918,43885,122010,340323,954394,
%T A065178 2685930,7588770,21507696,61144062,174283887,498012094,1426213191,
%U A065178 4092816966,11767176070,33890202192,97761428205,282424564744
%N A065178 Number of site swap patterns with 2 balls and exact period n.
%C A065178 When interspersed with 0's, exponents in expansion of A065481 as a product zeta(n)^(-a(n)).
%H A065178 G. C. Greubel, <a href="/A065178/b065178.txt">Table of n, a(n) for n = 1..1000</a>
%H A065178 Juggling Information Service, <a href="http://www.juggling.org/bin/mfs/JIS/help/siteswap/">Site Swap FAQs</a>
%H A065178 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, eq. (A1).
%H A065178 G. Niklasch, <a href="/A001692/a001692.html">Some number theoretical constants: 1000-digit values</a> [Cached copy]
%F A065178 a(n) ~ 3^n/n. - _Vaclav Kotesovec_, Mar 05 2016
%F A065178 Inverse Euler transform of A133494. - _Alois P. Heinz_, Jun 23 2018
%F A065178 G.f.: Sum_{k>=1} mu(k) * log(1 + x^k/(1 - 3*x^k))/k. - _Seiichi Manyama_, Apr 14 2025
%e A065178 We have one period 1 (2), two period 2 (31/13 and 40/04) and six period three 2-ball siteswaps (312, 330, 411, 420, 501, 600) (The average of the digits is always 2).
%p A065178 [seq(DistSS(p,2),p=1..60)];
%p A065178 A065178 := proc(n)
%p A065178     add( mobius(n/d)*(3^d-2^d),d=numtheory[divisors](n)) /n ;
%p A065178 end proc:
%p A065178 seq(A065178(n),n=1..30) ; # _R. J. Mathar_, Aug 05 2015
%t A065178 a[n_] := DivisorSum[n, MoebiusMu[n/#] * (3^#-2^#)&] / n; Array[a, 30] (* _Jean-François Alcover_, Mar 05 2016, after _R. J. Mathar_ *)
%Y A065178 Row 2 of A065177.
%Y A065178 Cf. A008683, A065179, A065180, A065481.
%K A065178 nonn
%O A065178 1,2
%A A065178 _Antti Karttunen_, Oct 19 2001