cp's OEIS Frontend

A020708 Pisot sequences E(4,9), P(4,9).

%I A020708 #33 Dec 19 2022 13:36:52
%S A020708 4,9,20,44,97,214,472,1041,2296,5064,11169,24634,54332,119833,264300,
%T A020708 582932,1285697,2835694,6254320,13794337,30424368,67103056,148000449,
%U A020708 326425266,719953588,1587907625,3502240516,7724434620,17036776865,37575794246,82876023112
%N A020708 Pisot sequences E(4,9), P(4,9).
%H A020708 Colin Barker, <a href="/A020708/b020708.txt">Table of n, a(n) for n = 0..1000</a>
%H A020708 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 A020708 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,1).
%F A020708 a(n) = 2*a(n-1) + a(n-3) (holds at least up to n = 1000 but is not known to hold in general).
%F A020708 Empirical g.f.: (4+x+2*x^2) / (1-2*x-x^3). - _Colin Barker_, Jun 05 2016
%F A020708 Theorem: E(4,9) satisfies a(n) = 2 a(n - 1) + 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
%t A020708 RecurrenceTable[{a[0] == 4, a[1] == 9, a[n] == Floor[a[n - 1]^2/a[n - 2] + 1/2]}, a, {n, 0, 30}] (* _Bruno Berselli_, Feb 05 2016 *)
%t A020708 LinearRecurrence[{2,0,1},{4,9,20},40] (* _Harvey P. Dale_, Dec 19 2022 *)
%o A020708 (Magma) Exy:=[4,9]; [n le 2 select Exy[n] else Floor(Self(n-1)^2/Self(n-2) + 1/2): n in [1..40]]; // _Bruno Berselli_, Feb 05 2016
%o A020708 (PARI) Vec((4+x+2*x^2) / (1-2*x-x^3) + O(x^30)) \\ _Jinyuan Wang_, Mar 10 2020
%Y A020708 This is a subsequence of A008998.
%Y A020708 See A008776 for definitions of Pisot sequences.
%K A020708 nonn
%O A020708 0,1
%A A020708 _David W. Wilson_