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

0,1

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).

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

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 *)

Prog

(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
(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

Crossrefs

This is a subsequence of A005314.

See A008776 for definitions of Pisot sequences.

Cf. A099529.

Sequence in context: A188555 A020958 A020750 * A090990 A225605 A088495

Adjacent sequences: A020710 A020711 A020712 * A020714 A020715 A020716

Keyword

nonn

Author

David W. Wilson

Status

approved