A279682 -id:A279682 - cp's OEIS frontend

A279673 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.

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

Offset: 0

Tanya Khovanova and Konstantin Knop, Dec 16 2016

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.

			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.

Colin Barker, Table of n, a(n) for n = 0..1000
Tanya Khovanova and Konstantin Knop, Coins that Change Their Weights, arXiv:1611.09201 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (2,-1,4).

Cf. A279674, A279682, A279684, A279685.

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

LinearRecurrence[{2, -1, 4}, {1, 3, 9}, 30]

Vec((1 + x + 4*x^2) / (1 - 2*x + x^2 - 4*x^3) + O(x^40)) \\ Colin Barker, Dec 17 2016

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

A279674 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.

Original entry on oeis.org

1, 3, 5, 11, 29, 67, 149, 347, 813, 1875, 4325, 10027, 23229, 53731, 124341, 287867, 666317, 1542131, 3569413, 8261963, 19123037, 44261763, 102448341, 237127067, 548852845, 1270371987, 2940399397, 6805838187, 15752764925, 36461289251, 84393166325, 195336103099

Offset: 0

Tanya Khovanova and Konstantin Knop, Dec 16 2016

An LHR-coin is a coin that can change its weight periodically from light to heavy to real to light.

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.

			If we have two weighings we are not allowed to have outcomes that consist of two imbalances. That means a(2) = 9 - 4 = 5.
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.

Colin Barker, Table of n, a(n) for n = 0..1000
Tanya Khovanova and Konstantin Knop, Coins that Change Their Weights, arXiv:1611.09201 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (2,-1,4).

Cf. A279673, A279682, A279684, A279685.

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

LinearRecurrence[{2, -1, 4}, {1, 3, 5}, 30]

Vec((1 + x) / (1 - 2*x + x^2 - 4*x^3) + O(x^40)) \\ Colin Barker, Dec 17 2016

a(n) = 2*a(n-1) - a(n-2) + 4*a(n-3).

G.f.: (1 + x) / (1 - 2*x + x^2 - 4*x^3). - Colin Barker, Dec 17 2016

A279684 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.

Original entry on oeis.org

1, 3, 5, 15, 37, 87, 205, 495, 1173, 2759, 6493, 15263, 35749, 83575, 195181, 455247, 1060533, 2468391, 5740925, 13342975, 30993349, 71956951, 166991501, 387397551, 898427605, 2083016071, 4828379549, 11189823071, 25928070117, 60069313847, 139148806829

Offset: 0

Tanya Khovanova and Konstantin Knop, Dec 16 2016

nonn

An LHR-coin is a coin that can change its weight periodically from light to heavy to real to light.

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.

			If we have two weighings we are not allowed to have outcomes that consist of two imbalances. That means a(2) = 9 - 4 = 5.
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.

Tanya Khovanova and Konstantin Knop, Coins that Change Their Weights, arXiv:1611.09201 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (3, -1, 1, -2, -8).

Cf. A279673, A279674, A279682, A279685.

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

LinearRecurrence[{3, -1, 1, -2, -8}, {1, 3, 5, 15, 37}, 30]

a(n) = 3a(n-1) - a(n-2) + a(n-3) - 2a(n-4) - 8a(n-5).

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

A279685 The maximum number of coins that can be processed in n weighings of an adaptive strategy that all are real (and identical) except for one LHR-coin starting in an unknown state.

Original entry on oeis.org

1, 1, 3, 6, 16, 39, 91, 216, 499, 1144, 2651, 6152, 14227, 32904, 76187, 176376, 408179, 944728, 2186779, 5061544, 11715219, 27116008, 62762971, 145270808, 336242675, 778266424, 1801373403, 4169451080, 9650594451, 22337231432, 51701672731

Offset: 0

Tanya Khovanova and Konstantin Knop, Dec 16 2016

An LHR-coin is a coin that can change its weight periodically from light to heavy to real to light.

Colin Barker, Table of n, a(n) for n = 0..1000
Tanya Khovanova and Konstantin Knop, Coins that Change Their Weights, arXiv:1611.09201 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (3,-3,5,-4).

Cf. A279673, A279674, A279682, A279684.

Vec((1 - 2*x + 3*x^2 - 5*x^3 + 6*x^4 - 2*x^5 + 4*x^6 + 4*x^7 - 7*x^8 - 4*x^9) / ((1 - x)*(1 - 2*x + x^2 - 4*x^3)) + O(x^40)) \\ Colin Barker, Dec 17 2016

a(n) = (A279673(n-1) + 1)/2 + A279673(n-2), for n > 6.
From Colin Barker, Dec 17 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + 5*a(n-3) - 4*a(n-4) for n>9.
G.f.: (1 - 2*x + 3*x^2 - 5*x^3 + 6*x^4 - 2*x^5 + 4*x^6 + 4*x^7 - 7*x^8 - 4*x^9) / ((1 - x)*(1 - 2*x + x^2 - 4*x^3)).
(End)

A293949 Arises in a coin-weighing problem.

Original entry on oeis.org

1, 1, 3, 6, 16, 41, 99, 235, 565, 1347, 3187, 7515, 17691, 41523, 97240, 227361, 530859, 1237881, 2883475, 6710758, 15606075, 36268307, 84239608, 195567275, 453832309, 1052784046, 2441461427, 5660381126, 13120263173, 30405643251, 70451852483, 163217723702, 378083386517

Offset: 0

N. J. A. Sloane, Oct 28 2017

nonn

The maximum number of coins that can be processed in n weighings where all coins are real except for one LHR-coin starting in an unknown state. - Robert Price, Nov 30 2017

Tanya Khovanova and Konstantin Knop, Coins that Change Their Weights, arXiv:1611.09201 [math.CO], 2016. See Section 7.

Cf. A279673, A279674, A279682, A279684.

a(8)-a(32) from Robert Price, Nov 30 2017

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A279673 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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A279674 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.

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A279684 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.

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A279685 The maximum number of coins that can be processed in n weighings of an adaptive strategy that all are real (and identical) except for one LHR-coin starting in an unknown state.

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A293949 Arises in a coin-weighing problem.

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