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

0,1

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).

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)

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 *)

Crossrefs

Cf. A077922.

Sequence in context: A256959 A181880 A291016 * A229242 A087449 A004253

Adjacent sequences: A010904 A010905 A010906 * A010908 A010909 A010910

Keyword

nonn

Author

Simon Plouffe

Status

approved