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 A020713 #35 Sep 08 2022 08:44:45
%S A020713 5,9,16,28,49,86,151,265,465,816,1432,2513,4410,7739,13581,23833,
%T A020713 41824,73396,128801,226030,396655,696081,1221537,2143648,3761840,
%U A020713 6601569,11584946,20330163,35676949,62608681,109870576,192809420,338356945,593775046,1042002567
%N A020713 Pisot sequences E(5,9), P(5,9).
%H A020713 Colin Barker, <a href="/A020713/b020713.txt">Table of n, a(n) for n = 0..1000</a>
%H A020713 Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger, <a href="https://arxiv.org/abs/1609.05570">Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences</a>, arXiv:1609.05570 [math.NT], 2016.
%H A020713 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1).
%F A020713 a(n) = 2*a(n-1) - a(n-2) + a(n-3) (holds at least up to n = 1000 but is not known to hold in general).
%F A020713 Empirical g.f.: (5-x+3*x^2) / (1-2*x+x^2-x^3). - _Colin Barker_, Jun 05 2016
%F A020713 Theorem: E(5,9) satisfies a(n) = 2 a(n - 1) - a(n - 2) + a(n - 3) for n>=3. Proved using the PtoRv program of Ekhad-Sloane-Zeilberger, and implies the above conjectures. - _N. J. A. Sloane_, Sep 09 2016
%F A020713 a(n) = (-1)^n * A099529(n+6). - _Jinyuan Wang_, Mar 10 2020
%t A020713 RecurrenceTable[{a[0] == 5, a[1] == 9, a[n] == Ceiling[a[n - 1]^2/a[n - 2]-1/2]}, a, {n, 0, 40}] (* _Bruno Berselli_, Feb 04 2016 *)
%t A020713 LinearRecurrence[{2,-1,1},{5,9,16},40] (* _Harvey P. Dale_, Aug 03 2021 *)
%o A020713 (Magma) Iv:=[5, 9]; [n le 2 select Iv[n] else Ceiling(Self(n-1)^2/Self(n-2)-1/2): n in [1..40]]; // _Bruno Berselli_, Feb 04 2016
%o A020713 (PARI) lista(nn) = {print1(x = 5, ", ", y = 9, ", "); for (n=1, nn, z = ceil(y^2/x -1/2); print1(z, ", "); x = y; y = z;);} \\ _Michel Marcus_, Feb 04 2016
%Y A020713 This is a subsequence of A005314.
%Y A020713 See A008776 for definitions of Pisot sequences.
%Y A020713 Cf. A099529.
%K A020713 nonn
%O A020713 0,1
%A A020713 _David W. Wilson_