OFFSET
0,2
COMMENTS
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 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.
The first seven terms coincide with sequence A102001, which counts all the outcomes that don't have three imbalances in a row.
This sequence also counts the possible outcomes starting in the light or heavy state, and for the coins starting in the real state the possible number of outcomes is a subset for coins starting in the light state.
LINKS
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,-1,1,-2,-8).
FORMULA
a(n) = 3*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) - 8*a(n-5).
G.f.: (1 + x^2 - 6*x^3)/((1 + x)*(1 - 2*x)*(1 - 2*x + x^2 - 4*x^3)). - Ilya Gutkovskiy, Dec 17 2016
EXAMPLE
Consider a(7): in addition to outcomes that do not have three imbalances in a row, we are not allowed to have any outcomes like <<=<=<<, in which the first (odd-numbered imbalance) and the fourth (even-numbered imbalance) are both followed by an imbalance. We can replace a less-than sign with a greater-than sign. That means a(7) = A102001(7) - 32 = 739 - 32 = 707.
MATHEMATICA
LinearRecurrence[{3, -1, 1, -2, -8}, {1, 3, 9, 19, 49}, 30]
PROG
(Magma) I:=[1, 3, 9, 19, 49]; [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..30]]; // Vincenzo Librandi, Dec 18 2016
(PARI) Vec((1 + x^2 - 6*x^3)/((1 + x)*(1 - 2*x)*(1 - 2*x + x^2 - 4*x^3)) + O(x^40)) \\ Colin Barker, Dec 19 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Tanya Khovanova and Konstantin Knop, Dec 16 2016
STATUS
approved