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 A020711 #57 Nov 01 2024 12:05:37
%S A020711 5,7,10,14,20,29,42,61,89,130,190,278,407,596,873,1279,1874,2746,4024,
%T A020711 5897,8642,12665,18561,27202,39866,58426,85627,125492,183917,269543,
%U A020711 395034,578950,848492,1243525,1822474,2670965,3914489,5736962,8407926,12322414,18059375
%N A020711 Pisot sequences E(5,7), P(5,7).
%H A020711 Colin Barker, <a href="/A020711/b020711.txt">Table of n, a(n) for n = 0..1000</a>
%H A020711 Andrei Asinowski, Cyril Banderier, Valerie Roitner, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/several_patterns.pdf">Generating functions for lattice paths with several forbidden patterns</a>, (2019).
%H A020711 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 A020711 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=911">Encyclopedia of Combinatorial Structures 911</a>
%H A020711 Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, <a href="https://ceur-ws.org/Vol-3792/paper19.pdf">Integer sequences from k-iterated line digraphs</a>, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
%H A020711 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-1).
%F A020711 a(n) = 2*a(n-1) - a(n-2) + a(n-3) - a(n-4) (holds at least up to n = 1000 but is not known to hold in general).
%F A020711 Empirical g.f.: -(4*x^3-x^2+3*x-5) / ((x-1)*(x^3+x-1)). - _Colin Barker_, Oct 07 2014
%F A020711 Theorem: E(5,7) satisfies a(n) = 3 a(n - 1) + 2 a(n - 2) + a(n - 3) - a(n - 4) for n >= 4. Proved using the PtoRv program of Ekhad-Sloane-Zeilberger, and implies the above conjectures. - _N. J. A. Sloane_, Sep 09 2016
%F A020711 Empirical formula: a(n) = a(n-1) + a(n-3) - 1. - _Greg Dresden_, May 18 2020
%t A020711 PSE[a_,b_,n_] := Join[{x = a, y = b}, Table[z = Floor[y^2/x + 1/2]; x = y; y = z, {n}]]; A020711 = PSE[5,7,50] (* _Vladimir Joseph Stephan Orlovsky_, Mar 26 2011 *)
%t A020711 LinearRecurrence[{2,-1,1,-1},{5,7,10,14},50] (* _Harvey P. Dale_, Jan 20 2017 *)
%o A020711 (PARI) Vec(-(4*x^3-x^2+3*x-5)/((x-1)*(x^3+x-1)) + O(x^40)) \\ _Jinyuan Wang_, Mar 10 2020
%Y A020711 See A008776 for definitions of Pisot sequences.
%K A020711 nonn
%O A020711 0,1
%A A020711 _David W. Wilson_