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%I #37 Sep 08 2022 08:44:45
%S 3,10,33,109,360,1189,3927,12970,42837,141481,467280,1543321,5097243,
%T 16835050,55602393,183642229,606529080,2003229469,6616217487,
%U 21851881930,72171863277,238367471761,787274278560,2600190307441,8587845200883,28363725910090
%N Pisot sequences E(3,10), P(3,10).
%H Colin Barker, <a href="/A020704/b020704.txt">Table of n, a(n) for n = 0..1000</a>
%H S. B. Ekhad, N. J. A. Sloane, D. Zeilberger, <a href="http://arxiv.org/abs/1609.05570">Automated proofs (or disproofs) of linear recurrences satisfied by Pisot Sequences</a>, arXiv:1609.05570 [math.NT] (2016)
%F a(n) = 3*a(n-1) + a(n-2) (holds at least up to n = 1000 but is not known to hold in general).
%F Conjectures from _Colin Barker_, Jun 05 2016: (Start)
%F a(n) = (2^(-1-n)*((3-sqrt(13))^n*(-11+3*sqrt(13)) + (3+sqrt(13))^n*(11+3*sqrt(13))))/sqrt(13).
%F G.f.: (3+x) / (1-3*x-x^2).
%F (End)
%F Theorem: For E(3,10), a(n) = 3 a(n - 1) + a(n - 2) for n>=2. 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] == 3, a[1] == 10, a[n] == Floor[a[n - 1]^2/a[n - 2] + 1/2]}, a, {n, 0, 30}] (* _Bruno Berselli_, Feb 05 2016 *)
%o (Magma) Exy:=[3,10]; [n le 2 select Exy[n] else Floor(Self(n-1)^2/Self(n-2) + 1/2): n in [1..30]]; // _Bruno Berselli_, Feb 05 2016
%Y This is a subsequence of A006190.
%Y See A008776 for definitions of Pisot sequences.
%K nonn
%O 0,1
%A _David W. Wilson_
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