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%I #20 Oct 24 2023 06:19:38
%S 1,3,18,105,606,3483,19986,114609,657054,3766515,21590418,123758649,
%T 709393758,4066287627,23308182162,133603718145,765823377054,
%U 4389738680547,25162205542674,144231041494281,826739659899486,4738913739770235,27163694339468562
%N Number of multiplex juggling sequences of length n, base state <1,1,1> and hand capacity 2.
%H Colin Barker, <a href="/A136779/b136779.txt">Table of n, a(n) for n = 1..1000</a>
%H Carolina Benedetti, Christopher R. H. Hanusa, Pamela E. Harris, Alejandro H. Morales, and Anthony Simpson, <a href="https://arxiv.org/abs/2001.03219">Kostant's partition function and magic multiplex juggling sequences</a>, arXiv:2001.03219 [math.CO], 2020. See Table 1 p. 12.
%H S. Butler and R. Graham, <a href="http://arXiv.org/abs/0801.2597">Enumerating (multiplex) juggling sequences</a>, arXiv:0801.2597 [math.CO], 2008.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-13).
%F G.f.: x*(1-5*x+7*x^2)/(1-8*x+13*x^2).
%F From _Colin Barker_, Aug 31 2016: (Start)
%F a(n) = 8*a(n-1)-13*a(n-2) for n>3.
%F a(n) = ((9-14*sqrt(3))*(4-sqrt(3))^n+(4+sqrt(3))^n*(9+14*sqrt(3)))/338 for n>1. (End)
%F E.g.f.: (91*x - 9 + exp(4*x)*(9*cosh(sqrt(3)*x) + 14*sqrt(3)*sinh(sqrt(3)*x)))/169. - _Stefano Spezia_, Oct 15 2023
%t LinearRecurrence[{8,-13},{1,3,18},25] (* _Paolo Xausa_, Oct 24 2023 *)
%o (PARI) Vec((x-5*x^2+7*x^3)/(1-8*x+13*x^2) + O(x^30)) \\ _Colin Barker_, Aug 31 2016
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 25); Coefficients(R!( (x-5*x^2+7*x^3)/(1-8*x+13*x^2))); // _Marius A. Burtea_, Jan 13 2020
%Y Cf. A136780.
%K nonn,easy
%O 1,2
%A _Steve Butler_, Jan 21 2008
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