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Pisot sequence E(3,11), a(n) = floor(a(n-1)^2/a(n-2) + 1/2).
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%I #37 Mar 12 2020 03:06:44

%S 3,11,40,145,526,1908,6921,25105,91065,330326,1198213,4346356,

%T 15765820,57188385,207443151,752472043,2729490816,9900859685,

%U 35914032730,130273308376,472548850273,1714107200301,6217692609825,22553841080350,81811015661001,296758421753528

%N Pisot sequence E(3,11), a(n) = floor(a(n-1)^2/a(n-2) + 1/2).

%H Colin Barker, <a href="/A010911/b010911.txt">Table of n, a(n) for n = 0..1000</a>

%H D. W. Boyd, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa34/aa3444.pdf">Some integer sequences related to the Pisot sequences</a>, Acta Arithmetica, 34 (1979), 295-305.

%H D. W. Boyd, <a href="https://www.researchgate.net/profile/David_Boyd7/publication/262181133_Linear_recurrence_relations_for_some_generalized_Pisot_sequences_-_annotated_with_corrections_and_additions/links/00b7d536d49781037f000000.pdf">Linear recurrence relations for some generalized Pisot sequences</a>, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.

%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).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,2,1).

%F Is it true that a(n+3)=3*a(n+2)+2*a(n+1)+a(n)? - Claude Lenormand (claude.lenormand(AT)free.fr), Dec 05 2001

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

%F Theorem: a(n) = 3 a(n - 1) + 2 a(n - 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

%F a(n) = A108153(n+2). - _Jinyuan Wang_, Mar 10 2020

%t LinearRecurrence[{3, 2, 1}, {3, 11, 40}, 30] (* _Jean-François Alcover_, Oct 05 2018 *)

%o (PARI) x='x+O('x^33); Vec((3+2*x+x^2)/(1-3*x-2*x^2-x^3)) \\ _Altug Alkan_, Oct 05 2018

%Y Cf. A108153.

%K nonn,easy

%O 0,1

%A _Simon Plouffe_