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%I #32 Jul 14 2023 14:48:19
%S 4,19,90,426,2016,9541,45154,213697,1011348,4786332,22651920,
%T 107203069,507352048,2401107571,11363544486,53779407822,254517831936,
%U 1204537747753,5700626846950,26978935702753,127681216679304,604267465267128,2859772009358880,13534231802298265,64052459384483260,303136344428812723,1434630991482656082,6789572149788327282
%N Pisot sequence E(4,19), a(n) = floor( a(n-1)^2/a(n-2)+1/2 ).
%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 Vincenzo Librandi, <a href="/A010907/b010907.txt">Table of n, a(n) for n = 0..200</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_04">Index entries for linear recurrences with constant coefficients</a>, signature (4, 3, 2, 1).
%F Theorem: a(n) = 4 a(n - 1) + 3 a(n - 2) + 2 a(n - 3) + a(n - 4). (Proved using the PtoRv program of Ekhad-Sloane-Zeilberger.) - _N. J. A. Sloane_, Sep 09 2016
%F G.f.: -(x^3+2*x^2+3*x+4)/(x^4+2*x^3+3*x^2+4*x-1). [_Colin Barker_, Nov 29 2012] (This follows from the above recurrence. - _N. J. A. Sloane_, Sep 09 2016)
%t PSE[a_,b_,n_]:=Join[{x=a,y=b}, Table[z=Floor[y^2/x+1/2]; x=y; y=z, {n}]]; A010907=PSE[4,19,20] (* _Zak Seidov_, Mar 24 2011 *)
%t nxt[{a_,b_}]:={b,Floor[b^2/a+1/2]}; Transpose[NestList[nxt,{4,19},20]] [[1]] (* _Harvey P. Dale_, Mar 13 2016 *)
%Y Cf. A077922.
%K nonn
%O 0,1
%A _Simon Plouffe_
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