%I #31 Jul 02 2023 13:46:04

%S 4,25,156,973,6069,37855,236118,1472770,9186303,57298942,357398265,

%T 2229247441,13904779737,86730120658,540973246008,3374283935917,

%U 21046867223484,131278407014845,818840161120305,5107459975407127,31857435234607602,198708591866413654

%N Pisot sequence E(4,25): a(n) = floor(a(n-1)^2/a(n-2)+1/2) for n>1, a(0)=4, a(1)=25.

%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="/A010909/b010909.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 (6, 1, 3).

%F Theorem: a(n) = 6 a(n - 1) + a(n - 2) + 3 a(n - 3). (Conjectured by _Colin Barker_, Jun 05 2016. Proved using the PtoRv program of Ekhad-Sloane-Zeilberger.) - _N. J. A. Sloane_, Sep 09 2016

%F G.f.: (4+x+2*x^2) / (1-6*x-x^2-3*x^3). (Follows from the recurrence.)

%t RecurrenceTable[{a[0] == 4, a[1] == 25, a[n] == Floor[a[n - 1]^2/a[n - 2] + 1/2]}, a, {n, 0, 25}] (* _Bruno Berselli_, Sep 03 2013 *)

%o (Magma) Exy:=[4,25]; [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

%K nonn

%O 0,1

%A _Simon Plouffe_

%E More terms from _Bruno Berselli_, Sep 03 2013