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%I #21 Mar 03 2024 22:38:39
%S 1,3,5,15,37,87,205,495,1173,2759,6493,15263,35749,83575,195181,
%T 455247,1060533,2468391,5740925,13342975,30993349,71956951,166991501,
%U 387397551,898427605,2083016071,4828379549,11189823071,25928070117,60069313847,139148806829
%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 or real 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 that start with a balance and every even-numbered imbalance that is not the last one must be followed by a balance, or every odd-numbered imbalance that is not the last one must be followed by a balance.
%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_05">Index entries for linear recurrences with constant coefficients</a>, signature (3, -1, 1, -2, -8).
%F a(n) = 3a(n-1) - a(n-2) + a(n-3) - 2a(n-4) - 8a(n-5).
%F G.f.: (1 - 3*x^2 + 2*x^3 - 4*x^4)/((1 + x)*(1 - 2*x)*(1 - 2*x + x^2 - 4*x^3)). - _Ilya Gutkovskiy_, 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 - 4 - 8 = 15.
%t LinearRecurrence[{3, -1, 1, -2, -8}, {1, 3, 5, 15, 37}, 30]
%o (Magma) I:=[1,3,5,15,37]; [n le 5 select I[n] else 3*Self(n-1)- Self(n-2)+Self(n-3)-2*Self(n-4)-8*Self(n-5): n in [1..40]]; // _Vincenzo Librandi_, Dec 16 2016
%Y Cf. A279673, A279674, A279682, A279685.
%K nonn
%O 0,2
%A _Tanya Khovanova_ and _Konstantin Knop_, Dec 16 2016