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

0,1

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

Sequence in context: A151330 A370570 A302871 * A142988 A356267 A056660

Adjacent sequences: A010910 A010911 A010912 * A010914 A010915 A010916

Keyword

nonn

Author

Simon Plouffe

Status

approved