OFFSET
0,1
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
S. B. Ekhad, N. J. A. Sloane, D. Zeilberger, Automated proofs (or disproofs) of linear recurrences satisfied by Pisot Sequences, arXiv:1609.05570 [math.NT] (2016)
FORMULA
a(n) = 3*a(n-1) + a(n-2) (holds at least up to n = 1000 but is not known to hold in general).
Conjectures from Colin Barker, Jun 05 2016: (Start)
a(n) = (2^(-1-n)*((3-sqrt(13))^n*(-11+3*sqrt(13)) + (3+sqrt(13))^n*(11+3*sqrt(13))))/sqrt(13).
G.f.: (3+x) / (1-3*x-x^2).
(End)
Theorem: For E(3,10), a(n) = 3 a(n - 1) + a(n - 2) for n>=2. 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] == 3, a[1] == 10, a[n] == Floor[a[n - 1]^2/a[n - 2] + 1/2]}, a, {n, 0, 30}] (* Bruno Berselli, Feb 05 2016 *)
PROG
(Magma) Exy:=[3, 10]; [n le 2 select Exy[n] else Floor(Self(n-1)^2/Self(n-2) + 1/2): n in [1..30]]; // Bruno Berselli, Feb 05 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved