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A010924 Pisot sequence E(8,55), a(n)=[ a(n-1)^2/a(n-2)+1/2 ]. 1

%I

%S 8,55,378,2598,17856,122724,843480,5797224,39844224,273848688,

%T 1882157472,12936036960,88909166592,611071221312,4199882327424,

%U 28865721292416,198393621719040,1363556058068736,9371698078726656,64411524820772352,442699337396994048

%N Pisot sequence E(8,55), a(n)=[ a(n-1)^2/a(n-2)+1/2 ].

%D D. W. Boyd, Some integer sequences related to the Pisot sequences, Acta Arithmetica, 34 (1979), 295-305.

%D Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger, Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences, Preprint, 2016.

%H Colin Barker, <a href="/A010924/b010924.txt">Table of n, a(n) for n = 0..1000</a>

%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 Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, <a href="http://www.emis.de/journals/JIS/VOL18/Szczyrba/sz3.html">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

%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 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6, 6).

%F Conjecture: a(n)=6*a(n-1)+6*a(n-2), n>1; a(0)=8, a(1)=55 . G.f.: (8+7x)/(1-6x-6x^2). - _Philippe Deléham_, Nov 19 2008

%F Theorem: a(n) = 6 a(n - 1) + 6 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 a[0] = 8; a[1] = 55; a[n_] := a[n] = Floor[a[n - 1]^2/a[n - 2] + 1/2]; Table[a[n], {n, 0, 20}] (* _Michael De Vlieger_, Jul 27 2016 *)

%o (PARI) pisotE(nmax, a1, a2) = {

%o a=vector(nmax); a[1]=a1; a[2]=a2;

%o for(n=3, nmax, a[n] = floor(a[n-1]^2/a[n-2]+1/2));

%o a

%o }

%o pisotE(50, 8, 55) \\ _Colin Barker_, Jul 27 2016

%K nonn,easy

%O 0,1

%A _Simon Plouffe_

%E "Index entries..." based on conjectured formula deleted by _Colin Barker_, Jul 27 2016

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Last modified May 26 19:20 EDT 2019. Contains 323597 sequences. (Running on oeis4.)