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A020721 Pisot sequences E(7,10), P(7,10).

%I A020721 #31 Sep 08 2022 08:44:45
%S A020721 7,10,14,20,29,42,61,89,130,190,278,407,596,873,1279,1874,2746,4024,
%T A020721 5897,8642,12665,18561,27202,39866,58426,85627,125492,183917,269543,
%U A020721 395034,578950,848492,1243525,1822474,2670965,3914489,5736962,8407926,12322414,18059375
%N A020721 Pisot sequences E(7,10), P(7,10).
%H A020721 Colin Barker, <a href="/A020721/b020721.txt">Table of n, a(n) for n = 0..1000</a>
%H A020721 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 A020721 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-1).
%F A020721 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 A020721 Empirical g.f.: (7-4*x+x^2-5*x^3) / ((1-x)*(1-x-x^3)). - _Colin Barker_, Jun 05 2016
%F A020721 Theorem: E(7,10) satisfies  a(n) = 2 a(n - 1) - a(n - 2) + a(n - 3) - a(n - 4) for n>=4. Proved using the PtoRv program of Ekhad-Sloane-Zeilberger. This shows that the above conjectures are correct. - _N. J. A. Sloane_, Sep 10 2016
%F A020721 a(n) = A020711(n+1). - _Jinyuan Wang_, Mar 10 2020
%F A020721 Empirical formula: a(n) = a(n-1) + a(n-3) - 1. - _Greg Dresden_, May 18 2020
%t A020721 RecurrenceTable[{a[0]==7, a[1]==10, a[n]== Floor[a[n-1]^2/a[n-2] +1/2]}, a, {n, 0, 50}] (* _Bruno Berselli_, Feb 05 2016 *)
%o A020721 (Magma) Exy:=[7,10]; [n le 2 select Exy[n] else Floor(Self(n-1)^2/Self(n-2) + 1/2): n in [1..50]]; // _Bruno Berselli_, Feb 05 2016
%o A020721 (PARI) Vec((7-4*x+x^2-5*x^3)/((1-x)*(1-x-x^3)) + O(x^40)) \\ _Jinyuan Wang_, Mar 10 2020
%Y A020721 Subsequence of A020711.
%Y A020721 See A008776 for definitions of Pisot sequences.
%Y A020721 Cf. A048626.
%K A020721 nonn
%O A020721 0,1
%A A020721 _David W. Wilson_