A020709 Pisot sequence E(4,10).

4, 10, 25, 63, 159, 401, 1011, 2549, 6427, 16205, 40859, 103021, 259755, 654941, 1651355, 4163693, 10498251, 26470077, 66741115, 168279693, 424297003, 1069813853, 2697406939, 6801187117, 17148375179, 43237565181, 109018319419, 274876578253, 693068226283

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 (3,-2,2).

Formula

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

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

Theorem: a(n) = 3 a(n - 1) - 2 a(n - 2) + 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

Mathematica

RecurrenceTable[{a[0] == 4, a[1] == 10, a[n] == Floor[a[n - 1]^2/a[n - 2] + 1/2]}, a, {n, 0, 40}] (* Bruno Berselli, Feb 05 2016 *)
LinearRecurrence[{3, -2, 2}, {4, 10, 25}, 30] (* Harvey P. Dale, Jan 29 2021 *)

Prog

(Magma) Exy:=[4, 10]; [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((4-2*x+3*x^2)/(1-3*x+2*x^2-2*x^3) + O(x^30)) \\ Jinyuan Wang, Mar 10 2020

Crossrefs

See A008776 for definitions of Pisot sequences.

Sequence in context: A033539 A020748 A021004 * A020734 A225301 A298412

Adjacent sequences: A020706 A020707 A020708 * A020710 A020711 A020712

Keyword

nonn

Author

David W. Wilson

Status

approved