A010907 Pisot sequence E(4,19), a(n) = floor( a(n-1)^2/a(n-2)+1/2 ).

4, 19, 90, 426, 2016, 9541, 45154, 213697, 1011348, 4786332, 22651920, 107203069, 507352048, 2401107571, 11363544486, 53779407822, 254517831936, 1204537747753, 5700626846950, 26978935702753, 127681216679304, 604267465267128, 2859772009358880, 13534231802298265, 64052459384483260, 303136344428812723, 1434630991482656082, 6789572149788327282

Offset: 0

Author

Simon Plouffe

Keywords

• nonn

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• D. W. Boyd, Some integer sequences related to the Pisot sequences, Acta Arithmetica, 34 (1979), 295-305
• D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.
• S. B. Ekhad, N. J. A. Sloane, D. Zeilberger, Automated proofs (or disproofs) of linear recurrences satisfied by Pisot Sequences, arXiv:1609.05570 [math.NT] (2016)
• Index entries for linear recurrences with constant coefficients, signature (4, 3, 2, 1).

Crossrefs

Cf. A077922.

Programs

Mathematica

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 *)
nxt[{a_,b_}]:={b,Floor[b^2/a+1/2]}; Transpose[NestList[nxt,{4,19},20]] [[1]] (* Harvey P. Dale, Mar 13 2016 *)

Formula

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

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)