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

Author

David W. Wilson

Keywords

• nonn

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
• Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
• Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-1).

Crossrefs

See A008776 for definitions of Pisot sequences.

Programs

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 *)

PARI

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

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