This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A279682 #26 Sep 08 2022 08:46:18
%S A279682 1,3,9,19,49,123,297,707,1697,4043,9561,22547,53073,124571,291721,
%T A279682 682083,1592577,3713643,8650425,20132275,46818225,108804923,252718825,
%U A279682 586701827,1361496929,3158352139,7324384281,16981143379,39360789521
%N A279682 The maximum number of coins that can be processed in n weighings where all coins are real except for one LHR-coin.
%C A279682 An LHR-coin is a coin that can change its weight periodically from light to heavy to real to light.
%C A279682 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.
%C A279682 The first seven terms coincide with sequence A102001, which counts all the outcomes that don't have three imbalances in a row.
%C A279682 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.
%H A279682 Colin Barker, <a href="/A279682/b279682.txt">Table of n, a(n) for n = 0..1000</a>
%H A279682 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 A279682 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,1,-2,-8).
%F A279682 a(n) = 3*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) - 8*a(n-5).
%F A279682 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
%e A279682 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.
%t A279682 LinearRecurrence[{3, -1, 1, -2, -8}, {1, 3, 9, 19, 49}, 30]
%o A279682 (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
%o A279682 (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
%Y A279682 Cf. A279673, A279674, A279684, A279685.
%K A279682 nonn,easy
%O A279682 0,2
%A A279682 _Tanya Khovanova_ and _Konstantin Knop_, Dec 16 2016