A010909 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.

4, 25, 156, 973, 6069, 37855, 236118, 1472770, 9186303, 57298942, 357398265, 2229247441, 13904779737, 86730120658, 540973246008, 3374283935917, 21046867223484, 131278407014845, 818840161120305, 5107459975407127, 31857435234607602, 198708591866413654

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

Colin Barker, Table of n, a(n) for n = 0..1000

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 (6, 1, 3).

Formula

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

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

Mathematica

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 *)

Prog

(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

Crossrefs

Sequence in context: A055846 A091634 A236580 * A079750 A195510 A264775

Adjacent sequences: A010906 A010907 A010908 * A010910 A010911 A010912

Keyword

nonn

Author

Simon Plouffe

Extensions

More terms from Bruno Berselli, Sep 03 2013

Status

approved