A020708 Pisot sequences E(4,9), P(4,9).

4, 9, 20, 44, 97, 214, 472, 1041, 2296, 5064, 11169, 24634, 54332, 119833, 264300, 582932, 1285697, 2835694, 6254320, 13794337, 30424368, 67103056, 148000449, 326425266, 719953588, 1587907625, 3502240516, 7724434620, 17036776865, 37575794246, 82876023112

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 (2,0,1).

Formula

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

Empirical g.f.: (4+x+2*x^2) / (1-2*x-x^3). - Colin Barker, Jun 05 2016

Theorem: E(4,9) satisfies a(n) = 2 a(n - 1) + 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

Mathematica

RecurrenceTable[{a[0] == 4, a[1] == 9, a[n] == Floor[a[n - 1]^2/a[n - 2] + 1/2]}, a, {n, 0, 30}] (* Bruno Berselli, Feb 05 2016 *)
LinearRecurrence[{2, 0, 1}, {4, 9, 20}, 40] (* Harvey P. Dale, Dec 19 2022 *)

Prog

(Magma) Exy:=[4, 9]; [n le 2 select Exy[n] else Floor(Self(n-1)^2/Self(n-2) + 1/2): n in [1..40]]; // Bruno Berselli, Feb 05 2016
(PARI) Vec((4+x+2*x^2) / (1-2*x-x^3) + O(x^30)) \\ Jinyuan Wang, Mar 10 2020

Crossrefs

This is a subsequence of A008998.

See A008776 for definitions of Pisot sequences.

Sequence in context: A132175 A019493 A019492 * A345192 A144685 A109110

Adjacent sequences: A020705 A020706 A020707 * A020709 A020710 A020711

Keyword

nonn

Author

David W. Wilson

Status

approved