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Pisot sequences E(5,7), P(5,7).
3

%I #53 May 21 2020 10:26:15

%S 5,7,10,14,20,29,42,61,89,130,190,278,407,596,873,1279,1874,2746,4024,

%T 5897,8642,12665,18561,27202,39866,58426,85627,125492,183917,269543,

%U 395034,578950,848492,1243525,1822474,2670965,3914489,5736962,8407926,12322414,18059375

%N Pisot sequences E(5,7), P(5,7).

%H Colin Barker, <a href="/A020711/b020711.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 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 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=911">Encyclopedia of Combinatorial Structures 911</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-1).

%F 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 Empirical g.f.: -(4*x^3-x^2+3*x-5) / ((x-1)*(x^3+x-1)). - _Colin Barker_, Oct 07 2014

%F 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 Empirical formula: a(n) = a(n-1) + a(n-3) - 1. - _Greg Dresden_, May 18 2020

%t 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 LinearRecurrence[{2,-1,1,-1},{5,7,10,14},50] (* _Harvey P. Dale_, Jan 20 2017 *)

%o (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 See A008776 for definitions of Pisot sequences.

%K nonn

%O 0,1

%A _David W. Wilson_