A020713 Pisot sequences E(5,9), P(5,9).

5, 9, 16, 28, 49, 86, 151, 265, 465, 816, 1432, 2513, 4410, 7739, 13581, 23833, 41824, 73396, 128801, 226030, 396655, 696081, 1221537, 2143648, 3761840, 6601569, 11584946, 20330163, 35676949, 62608681, 109870576, 192809420, 338356945, 593775046, 1042002567

Offset: 0

Author

David W. Wilson

Keywords

• nonn

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger, Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences, arXiv:1609.05570 [math.NT], 2016.
• Index entries for linear recurrences with constant coefficients, signature (2,-1,1).

Crossrefs

This is a subsequence of A005314.

See A008776 for definitions of Pisot sequences.

Cf. A099529.

Programs

Magma

Iv:=[5, 9]; [n le 2 select Iv[n] else Ceiling(Self(n-1)^2/Self(n-2)-1/2): n in [1..40]]; // Bruno Berselli, Feb 04 2016

Mathematica

RecurrenceTable[{a[0] == 5, a[1] == 9, a[n] == Ceiling[a[n - 1]^2/a[n - 2]-1/2]}, a, {n, 0, 40}] (* Bruno Berselli, Feb 04 2016 *)
LinearRecurrence[{2,-1,1},{5,9,16},40] (* Harvey P. Dale, Aug 03 2021 *)

PARI

lista(nn) = {print1(x = 5, ", ", y = 9, ", "); for (n=1, nn, z = ceil(y^2/x -1/2); print1(z, ", "); x = y; y = z;);} \\ Michel Marcus, Feb 04 2016

Formula

a(n) = 2*a(n-1) - a(n-2) + a(n-3) (holds at least up to n = 1000 but is not known to hold in general).

Empirical g.f.: (5-x+3*x^2) / (1-2*x+x^2-x^3). - Colin Barker, Jun 05 2016

Theorem: E(5,9) satisfies a(n) = 2 a(n - 1) - a(n - 2) + a(n - 3) for n>=3. Proved using the PtoRv program of Ekhad-Sloane-Zeilberger, and implies the above conjectures. - N. J. A. Sloane, Sep 09 2016

a(n) = (-1)^n * A099529(n+6). - Jinyuan Wang, Mar 10 2020