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
KEYWORD
nonn
AUTHOR
STATUS
approved