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.
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 (6,-2).
FORMULA
Up to n=10^5, a(n) = 6a(n-1) - 2a(n-2). - Ralf Stephan, Sep 03 2013
Conjecture: If p[i]=fibonacci(2i+2) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= det A. - Milan Janjic, May 08 2010
Conjectures from Colin Barker, Jun 05 2016: (Start)
a(n) = (((3-sqrt(7))^n*(-8+3*sqrt(7))+(3+sqrt(7))^n*(8+3*sqrt(7))))/(2*sqrt(7)).
a(n) = 6*a(n-1)-2*a(n-2) for n>1.
G.f.: (3-x) / (1-6*x+2*x^2). (End)
Theorem: a(n) = 6 a(n - 1) - 2 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[1] == 3, a[2] == 17, a[n] == Floor[a[n-1]^2/a[n-2]+1/2]}, a, {n, 40}] (* Vincenzo Librandi, Aug 09 2016 *)
LinearRecurrence[{6, -2}, {3, 17}, 30] (* Harvey P. Dale, Nov 12 2021 *)
PROG
(PARI) Vec((3-x)/(1-6*x+2*x^2) + O(x^25)) \\ Jinyuan Wang, Mar 10 2020~
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved