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Pisot sequence E(3,17), a(n) = floor( a(n-1)^2/a(n-2)+1/2 ).
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%I #39 Nov 12 2021 16:14:17

%S 3,17,96,542,3060,17276,97536,550664,3108912,17552144,99095040,

%T 559465952,3158605632,17832701888,100679000064,568408596608,

%U 3209093579520,18117744283904,102288278544384,577494182698496,3260388539102208,18407342869216256,103923280137093120

%N Pisot sequence E(3,17), a(n) = floor( a(n-1)^2/a(n-2)+1/2 ).

%H Colin Barker, <a href="/A010913/b010913.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 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_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-2).

%F Up to n=10^5, a(n) = 6a(n-1) - 2a(n-2). - _Ralf Stephan_, Sep 03 2013

%F Conjecture: If p[i]=fibonacci(2i+2) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= det A. - _Milan Janjic_, May 08 2010

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

%F a(n) = (((3-sqrt(7))^n*(-8+3*sqrt(7))+(3+sqrt(7))^n*(8+3*sqrt(7))))/(2*sqrt(7)).

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

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

%F Theorem: a(n) = 6 a(n - 1) - 2 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[1] == 3, a[2] == 17, a[n] == Floor[a[n-1]^2/a[n-2]+1/2]}, a, {n, 40}] (* _Vincenzo Librandi_, Aug 09 2016 *)

%t LinearRecurrence[{6,-2},{3,17},30] (* _Harvey P. Dale_, Nov 12 2021 *)

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

%K nonn

%O 0,1

%A _Simon Plouffe_