A020716 Pisot sequences E(6,8), P(6,8).

6, 8, 11, 15, 20, 27, 36, 48, 64, 85, 113, 150, 199, 264, 350, 464, 615, 815, 1080, 1431, 1896, 2512, 3328, 4409, 5841, 7738, 10251, 13580, 17990, 23832, 31571, 41823, 55404, 73395, 97228, 128800, 170624, 226029, 299425, 396654, 525455, 696080, 922110, 1221536

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 (1,1,0,-1).

Crossrefs

This is a subsequence of A023434.

See A008776 for definitions of Pisot sequences.

Programs

Magma

Exy:=[6,8]; [n le 2 select Exy[n] else Floor(Self(n-1)^2/Self(n-2) + 1/2): n in [1..50]]; // Bruno Berselli, Feb 05 2016

Mathematica

RecurrenceTable[{a[0]==6, a[1]==8, a[n]== Floor[a[n-1]^2/a[n-2] + 1/2]}, a, {n, 0, 50}] (* Bruno Berselli, Feb 05 2016 *)
LinearRecurrence[{1,1,0,-1},{6,8,11,15},50] (* Harvey P. Dale, Jul 27 2025 *)

PARI

Vec((6+2*x-3*x^2-4*x^3)/((1-x)*(1-x^2-x^3)) + O(x^50)) \\ Jinyuan Wang, Mar 10 2020

Formula

a(n) = a(n-1) + a(n-2) - a(n-4) (holds at least up to n = 1000 but is not known to hold in general).

Empirical g.f.: (6+2*x-3*x^2-4*x^3) / ((1-x)*(1-x^2-x^3)). - Colin Barker, Jun 05 2016

Theorem: E(6,8) satisfies a(n) = a(n - 1) + a(n - 2) - a(n - 4) for n>=4. Proved using the PtoRv program of Ekhad-Sloane-Zeilberger. This shows that the above conjectures are correct. - N. J. A. Sloane, Sep 10 2016

a(n) = a(n-2) + a(n-3) + 1. - Greg Dresden, May 18 2020