A020711 Pisot sequences E(5,7), P(5,7).

5, 7, 10, 14, 20, 29, 42, 61, 89, 130, 190, 278, 407, 596, 873, 1279, 1874, 2746, 4024, 5897, 8642, 12665, 18561, 27202, 39866, 58426, 85627, 125492, 183917, 269543, 395034, 578950, 848492, 1243525, 1822474, 2670965, 3914489, 5736962, 8407926, 12322414, 18059375

Offset

0,1

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Andrei Asinowski, Cyril Banderier, Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).

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.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 911

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-1).

Formula

a(n) = 2*a(n-1) - a(n-2) + a(n-3) - a(n-4) (holds at least up to n = 1000 but is not known to hold in general).

Empirical g.f.: -(4*x^3-x^2+3*x-5) / ((x-1)*(x^3+x-1)). - Colin Barker, Oct 07 2014

Theorem: E(5,7) satisfies a(n) = 3 a(n - 1) + 2 a(n - 2) + a(n - 3) - a(n - 4) for n >= 4. Proved using the PtoRv program of Ekhad-Sloane-Zeilberger, and implies the above conjectures. - N. J. A. Sloane, Sep 09 2016

Empirical formula: a(n) = a(n-1) + a(n-3) - 1. - Greg Dresden, May 18 2020

Mathematica

PSE[a_, b_, n_] := Join[{x = a, y = b}, Table[z = Floor[y^2/x + 1/2]; x = y; y = z, {n}]]; A020711 = PSE[5, 7, 50] (* Vladimir Joseph Stephan Orlovsky, Mar 26 2011 *)
LinearRecurrence[{2, -1, 1, -1}, {5, 7, 10, 14}, 50] (* Harvey P. Dale, Jan 20 2017 *)

Prog

(PARI) Vec(-(4*x^3-x^2+3*x-5)/((x-1)*(x^3+x-1)) + O(x^40)) \\ Jinyuan Wang, Mar 10 2020

Crossrefs

See A008776 for definitions of Pisot sequences.

Sequence in context: A071911 A070875 A091522 * A183044 A133756 A196936

Adjacent sequences: A020708 A020709 A020710 * A020712 A020713 A020714

Keyword

nonn

Author

David W. Wilson

Status

approved