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Pisot sequence E(4,10).
1

%I #26 Sep 08 2022 08:44:45

%S 4,10,25,63,159,401,1011,2549,6427,16205,40859,103021,259755,654941,

%T 1651355,4163693,10498251,26470077,66741115,168279693,424297003,

%U 1069813853,2697406939,6801187117,17148375179,43237565181,109018319419,274876578253,693068226283

%N Pisot sequence E(4,10).

%H Colin Barker, <a href="/A020709/b020709.txt">Table of n, a(n) for n = 0..1000</a>

%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 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,2).

%F a(n) = 3*a(n-1) - 2*a(n-2) + 2*a(n-3) (holds at least up to n = 1000 but is not known to hold in general).

%F Empirical g.f.: (4-2*x+3*x^2) / (1-3*x+2*x^2-2*x^3). - _Colin Barker_, Jun 05 2016

%F Theorem: a(n) = 3 a(n - 1) - 2 a(n - 2) + 2 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 RecurrenceTable[{a[0] == 4, a[1] == 10, a[n] == Floor[a[n - 1]^2/a[n - 2] + 1/2]}, a, {n, 0, 40}] (* _Bruno Berselli_, Feb 05 2016 *)

%t LinearRecurrence[{3,-2,2},{4,10,25},30] (* _Harvey P. Dale_, Jan 29 2021 *)

%o (Magma) Exy:=[4,10]; [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 (PARI) Vec((4-2*x+3*x^2)/(1-3*x+2*x^2-2*x^3) + O(x^30)) \\ _Jinyuan Wang_, Mar 10 2020

%Y See A008776 for definitions of Pisot sequences.

%K nonn

%O 0,1

%A _David W. Wilson_