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%I #22 Sep 08 2022 08:46:18
%S 1,3,5,11,29,67,149,347,813,1875,4325,10027,23229,53731,124341,287867,
%T 666317,1542131,3569413,8261963,19123037,44261763,102448341,237127067,
%U 548852845,1270371987,2940399397,6805838187,15752764925,36461289251,84393166325,195336103099
%N The maximum number of coins that can be processed in n weighings that all are real except for one LHR-coin starting in the heavy state.
%C An LHR-coin is a coin that can change its weight periodically from light to heavy to real to light.
%C Also the number of outcomes of n weighings such that every odd-numbered imbalance that is not the last one must be followed by a balance.
%H Colin Barker, <a href="/A279674/b279674.txt">Table of n, a(n) for n = 0..1000</a>
%H Tanya Khovanova and Konstantin Knop, <a href="https://arxiv.org/abs/1611.09201">Coins that Change Their Weights</a>, arXiv:1611.09201 [math.CO], 2016.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,4).
%F a(n) = 2*a(n-1) - a(n-2) + 4*a(n-3).
%F G.f.: (1 + x) / (1 - 2*x + x^2 - 4*x^3). - _Colin Barker_, Dec 17 2016
%e If we have two weighings we are not allowed to have outcomes that consist of two imbalances. That means a(2) = 9 - 4 = 5.
%e If we have three weighings we are not allowed the following outcomes: =<<, <<=, <<<, where any less-than sign can be interchanged with a greater-than sign. Thus a(3) = 27 - 2*4 - 8 = 11.
%t LinearRecurrence[{2, -1, 4}, {1, 3, 5}, 30]
%o (Magma) I:=[1,3,5]; [n le 3 select I[n] else 2*Self(n-1)-Self(n-2)+4*Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Dec 17 2016
%o (PARI) Vec((1 + x) / (1 - 2*x + x^2 - 4*x^3) + O(x^40)) \\ _Colin Barker_, Dec 17 2016
%Y Cf. A279673, A279682, A279684, A279685.
%K nonn,easy
%O 0,2
%A _Tanya Khovanova_ and _Konstantin Knop_, Dec 16 2016