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 (1,1,0,-1).
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
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved