A020721 Pisot sequences E(7,10), P(7,10).

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

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 (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.: (7-4*x+x^2-5*x^3) / ((1-x)*(1-x-x^3)). - Colin Barker, Jun 05 2016

Theorem: E(7,10) satisfies a(n) = 2 a(n - 1) - a(n - 2) + a(n - 3) - a(n - 4) for n>=4. Proved using the PtoRv program of Ekhad-Sloane-Zeilberger. This shows that the above conjectures are correct. - N. J. A. Sloane, Sep 10 2016

a(n) = A020711(n+1). - Jinyuan Wang, Mar 10 2020

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

Mathematica

RecurrenceTable[{a[0]==7, a[1]==10, a[n]== Floor[a[n-1]^2/a[n-2] +1/2]}, a, {n, 0, 50}] (* Bruno Berselli, Feb 05 2016 *)

Prog

(Magma) Exy:=[7, 10]; [n le 2 select Exy[n] else Floor(Self(n-1)^2/Self(n-2) + 1/2): n in [1..50]]; // Bruno Berselli, Feb 05 2016
(PARI) Vec((7-4*x+x^2-5*x^3)/((1-x)*(1-x-x^3)) + O(x^40)) \\ Jinyuan Wang, Mar 10 2020

Crossrefs

Subsequence of A020711.

See A008776 for definitions of Pisot sequences.

Cf. A048626.

Sequence in context: A134302 A229306 A023485 * A015782 A336089 A154681

Adjacent sequences: A020718 A020719 A020720 * A020722 A020723 A020724

Keyword

nonn

Author

David W. Wilson

Status

approved