OFFSET
0,1
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
D. W. Boyd, Some integer sequences related to the Pisot sequences, Acta Arithmetica, 34 (1979), 295-305.
D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.
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.
Index entries for linear recurrences with constant coefficients, signature (6,5).
FORMULA
Conjectures from Colin Barker, Jun 05 2016: (Start)
a(n) = (((3-sqrt(14))^n*(-15+4*sqrt(14))+(3+sqrt(14))^n*(15+4*sqrt(14))))/(2*sqrt(14)).
a(n) = 6*a(n-1)+5*a(n-2) for n>1.
G.f.: (4+3*x) / (1-6*x-5*x^2).
(End)
Theorem: a(n) = 6 a(n - 1) + 5 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] == 4, a[1] == 27, a[n] == Floor[a[n - 1]^2/a[n - 2] + 1/2]}, a, {n, 0, 25}] (* Bruno Berselli, Sep 03 2013 *)
nxt[{a_, b_}]:={b, Floor[b^2/a+1/2]}; NestList[nxt, {4, 27}, 30][[All, 1]] (* Harvey P. Dale, May 13 2018 *)
PROG
(Magma) Exy:=[4, 27]; [n le 2 select Exy[n] else Floor(Self(n-1)^2/Self(n-2)+1/2): n in [1..25]]; // Bruno Berselli, Sep 03 2013
(PARI) Vec((4+3*x)/(1-6*x-5*x^2) + O(x^25)) \\ Jinyuan Wang, Mar 10 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Bruno Berselli, Sep 03 2013
STATUS
approved