A010913 Pisot sequence E(3,17), a(n) = floor( a(n-1)^2/a(n-2)+1/2 ).

3, 17, 96, 542, 3060, 17276, 97536, 550664, 3108912, 17552144, 99095040, 559465952, 3158605632, 17832701888, 100679000064, 568408596608, 3209093579520, 18117744283904, 102288278544384, 577494182698496, 3260388539102208, 18407342869216256, 103923280137093120

Offset: 0

Author

Simon Plouffe

Keywords

• nonn

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).

Programs

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 *)

PARI

Vec((3-x)/(1-6*x+2*x^2) + O(x^25)) \\ Jinyuan Wang, Mar 10 2020~

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