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Pisot sequence E(4,27): a(n) = floor(a(n-1)^2/a(n-2)+1/2) for n>1, a(0)=4, a(1)=27.
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%I #34 Sep 08 2022 08:44:37

%S 4,27,182,1227,8272,55767,375962,2534607,17087452,115197747,776623742,

%T 5235731187,35297505832,237963690927,1604269674722,10815436502967,

%U 72913967391412,491560986863307,3313935758136902,22341419483137947,150618195689512192,1015416271552762887

%N Pisot sequence E(4,27): a(n) = floor(a(n-1)^2/a(n-2)+1/2) for n>1, a(0)=4, a(1)=27.

%H Colin Barker, <a href="/A010910/b010910.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_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,5).

%F Conjectures from _Colin Barker_, Jun 05 2016: (Start)

%F a(n) = (((3-sqrt(14))^n*(-15+4*sqrt(14))+(3+sqrt(14))^n*(15+4*sqrt(14))))/(2*sqrt(14)).

%F a(n) = 6*a(n-1)+5*a(n-2) for n>1.

%F G.f.: (4+3*x) / (1-6*x-5*x^2).

%F (End)

%F Theorem: a(n) = 6 a(n - 1) + 5 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] == 4, a[1] == 27, a[n] == Floor[a[n - 1]^2/a[n - 2] + 1/2]}, a, {n, 0, 25}] (* _Bruno Berselli_, Sep 03 2013 *)

%t nxt[{a_,b_}]:={b,Floor[b^2/a+1/2]}; NestList[nxt,{4,27},30][[All,1]] (* _Harvey P. Dale_, May 13 2018 *)

%o (Magma) Exy:=[4,27]; [n le 2 select Exy[n] else Floor(Self(n-1)^2/Self(n-2)+1/2): n in [1..25]]; // _Bruno Berselli_, Sep 03 2013

%o (PARI) Vec((4+3*x)/(1-6*x-5*x^2) + O(x^25)) \\ _Jinyuan Wang_, Mar 10 2020

%K nonn

%O 0,1

%A _Simon Plouffe_

%E More terms from _Bruno Berselli_, Sep 03 2013