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
Keywords
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).
Programs
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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
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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 *) -
PARI
Vec((4+x+2*x^2) / (1-2*x-x^3) + O(x^30)) \\ Jinyuan Wang, Mar 10 2020
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