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A279673
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The maximum number of coins that can be processed in n weighings where all coins are real except for one LHR-coin starting in the light state.
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6
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1, 3, 9, 19, 41, 99, 233, 531, 1225, 2851, 6601, 15251, 35305, 81763, 189225, 437907, 1013641, 2346275, 5430537, 12569363, 29093289, 67339363, 155862889, 360759571, 835013705, 1932719395, 4473463369, 10354262163, 23965938537, 55471468387, 128394046889
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OFFSET
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0,2
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COMMENTS
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An LHR-coin is a coin that can change its weight periodically from light to heavy to real to light.
If an LHR-coin starts in the real state, then the maximum number of coins that can be processed in n weighings is a(n-1).
Also the number of outcomes of n weighings such that every even-numbered imbalance that is not the last one must be followed by a balance.
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LINKS
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FORMULA
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a(n) = 2a(n-1) - a(n-2) + 4a(n-3).
G.f.: (1 + x + 4*x^2) / (1 - 2*x + x^2 - 4*x^3). - Colin Barker, Dec 17 2016
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EXAMPLE
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If we have three weighings we are not allowed to have outcomes that consist of three imbalances. That means a(3) = 27 - 8 = 19.
If we have four weighings we are not allowed the following outcomes: =<<<, <=<<, <<<=, <<<<, where any less-than sign can be interchanged with a greater-than sign. Thus a(4) = 81 - 3*8 - 16 = 41.
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MATHEMATICA
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LinearRecurrence[{2, -1, 4}, {1, 3, 9}, 30]
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PROG
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(Magma) I:=[1, 3, 9]; [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
(PARI) Vec((1 + x + 4*x^2) / (1 - 2*x + x^2 - 4*x^3) + O(x^40)) \\ Colin Barker, Dec 17 2016
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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