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
KEYWORD
nonn
AUTHOR
STATUS
approved