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A020716 Pisot sequences E(6,8), P(6,8).

%I A020716 #34 Jul 27 2025 16:13:37
%S A020716 6,8,11,15,20,27,36,48,64,85,113,150,199,264,350,464,615,815,1080,
%T A020716 1431,1896,2512,3328,4409,5841,7738,10251,13580,17990,23832,31571,
%U A020716 41823,55404,73395,97228,128800,170624,226029,299425,396654,525455,696080,922110,1221536
%N A020716 Pisot sequences E(6,8), P(6,8).
%H A020716 Colin Barker, <a href="/A020716/b020716.txt">Table of n, a(n) for n = 0..1000</a>
%H A020716 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 A020716 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1).
%F A020716 a(n) = a(n-1) + a(n-2) - a(n-4) (holds at least up to n = 1000 but is not known to hold in general).
%F A020716 Empirical g.f.: (6+2*x-3*x^2-4*x^3) / ((1-x)*(1-x^2-x^3)). - _Colin Barker_, Jun 05 2016
%F A020716 Theorem: E(6,8) satisfies a(n) = a(n - 1) + a(n - 2) - 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 A020716 a(n) = a(n-2) + a(n-3) + 1. - _Greg Dresden_, May 18 2020
%t A020716 RecurrenceTable[{a[0]==6, a[1]==8, a[n]== Floor[a[n-1]^2/a[n-2] + 1/2]}, a, {n, 0, 50}] (* _Bruno Berselli_, Feb 05 2016 *)
%t A020716 LinearRecurrence[{1,1,0,-1},{6,8,11,15},50] (* _Harvey P. Dale_, Jul 27 2025 *)
%o A020716 (Magma) Exy:=[6,8]; [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 A020716 (PARI) Vec((6+2*x-3*x^2-4*x^3)/((1-x)*(1-x^2-x^3)) + O(x^50)) \\ _Jinyuan Wang_, Mar 10 2020
%Y A020716 This is a subsequence of A023434.
%Y A020716 See A008776 for definitions of Pisot sequences.
%K A020716 nonn
%O A020716 0,1
%A A020716 _David W. Wilson_