A020711 -id:A020711 - cp's OEIS frontend

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

David W. Wilson

nonn

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).

Subsequence of A020711.
See A008776 for definitions of Pisot sequences.
Cf. A048626.

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

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 *)

Vec((7-4*x+x^2-5*x^3)/((1-x)*(1-x-x^3)) + O(x^40)) \\ Jinyuan Wang, Mar 10 2020

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

A128424 a(n) = floor(sqrt(a(n-1)^2 + a(n-2)^2 + a(n-1)*a(n-2))), a(1)=1, a(2)=3.

Original entry on oeis.org

1, 3, 3, 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

Offset: 1

Zak Seidov, May 04 2007

nonn

For a triangle with sides a(n-1) and a(n-2) and a 120-degree angle between them, a(n) is the floor of the value of the third side.

a(n) = A020711(n-4) for 4 <= n <= 41. - Georg Fischer, Nov 02 2018

Cf. A020711, A104803, A104804, A104805.

a[1]=1;a[2]=3;a[n_]:=a[n]=Floor[Sqrt[a[n-1]^2+a[n-2]^2+a[n-1]*a[n-2]]] Table[a[n],{n,45}]
RecurrenceTable[{a[1]==1,a[2]==3,a[n]==Floor[Sqrt[a[n-1]^2+a[n-2]^2+ a[n-1]*a[n-2]]]},a,{n,50}] (* Harvey P. Dale, Oct 01 2018 *)

Conjectures from Colin Barker, Nov 03 2018: (Start)
G.f.: x*(1 + x - 2*x^2 + x^3 - 2*x^4 + x^5 - x^6) / ((1 - x)*(1 - x - x^3)).
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - a(n-4) for n>7.
(End)

cp's OEIS Frontend

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

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