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 (2,-1,1).
FORMULA
a(n) = 2*a(n-1) - a(n-2) + a(n-3) (holds at least up to n = 1000 but is not known to hold in general).
Empirical g.f.: (5-x+3*x^2) / (1-2*x+x^2-x^3). - Colin Barker, Jun 05 2016
Theorem: E(5,9) satisfies a(n) = 2 a(n - 1) - a(n - 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
a(n) = (-1)^n * A099529(n+6). - Jinyuan Wang, Mar 10 2020
MATHEMATICA
RecurrenceTable[{a[0] == 5, a[1] == 9, a[n] == Ceiling[a[n - 1]^2/a[n - 2]-1/2]}, a, {n, 0, 40}] (* Bruno Berselli, Feb 04 2016 *)
LinearRecurrence[{2, -1, 1}, {5, 9, 16}, 40] (* Harvey P. Dale, Aug 03 2021 *)
PROG
(Magma) Iv:=[5, 9]; [n le 2 select Iv[n] else Ceiling(Self(n-1)^2/Self(n-2)-1/2): n in [1..40]]; // Bruno Berselli, Feb 04 2016
(PARI) lista(nn) = {print1(x = 5, ", ", y = 9, ", "); for (n=1, nn, z = ceil(y^2/x -1/2); print1(z, ", "); x = y; y = z; ); } \\ Michel Marcus, Feb 04 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved