Lyness cycles

2017 ◽  
Vol 101 (551) ◽  
pp. 193-207 ◽  
Author(s):  
Stan Dolan
Keyword(s):  
Elliptic Curves ◽  
Recurrence Relation ◽  
Recurrence Relations ◽  
Initial Values ◽  
Image Position ◽  
Almost All

In 1942, R. C. Lyness noted that some recurrence relations generate cycles, irrespective of the initial values. For example, the order 2 recurrence relationgenerates a cycle of period 5 for almost all values of u1 and u2 [1].The globally periodic nature of sequences generated by this recurrence relation can be seen by setting u1 = x and u2 = y. The sequence is thenLyness gave other examples of such recurrence relations but had been unable to find one with period 7 and challenged readers of the Gazette to find such a recurrence relation or prove it to be impossible.No answer to this challenge was forthcoming. However, since Lyness's time, interest in these cycles has been maintained due to links with cross-ratios and elliptic curves. In recent years, Jonny Griffiths has done much to popularise these cycles [2].

The geometric unfolding of recurrence relations

2020 ◽  
Vol 104 (561) ◽  
pp. 403-411
Author(s):  
Stan Dolan
Keyword(s):  
Recurrence Relation ◽  
Recurrence Relations ◽  
Initial Values ◽  
Almost All

In 1942 R. C. Lyness challenged readers of the Gazette to find a recurrence relation of order 2 which would generate a cycle of period 7 for almost all initial values [1].

scholarly journals On linear recurrence sequences with polynomial coefficients

1996 ◽  
Vol 38 (2) ◽  
pp. 147-155 ◽  
Author(s):  
A. J. van der Poorten ◽  
I. E. Shparlinski
Keyword(s):  
Recurrence Relation ◽  
Upper Bound ◽  
Rational Numbers ◽  
Negative Integer ◽  
Linear Recurrence ◽  
Initial Values ◽  
Polynomial Coefficients ◽  
Image Position ◽  
Recurrence Sequences

We consider sequences (Ah)defined over the field ℚ of rational numbers and satisfying a linear homogeneous recurrence relationwith polynomial coefficients sj;. We shall assume without loss of generality, as we may, that the sj, are defined over ℤ and the initial values A0A]…, An−1 are integer numbers. Also, without loss of generality we may assume that S0 and Sn have no non-negative integer zero. Indeed, any other case can be reduced to this one by making a shift h → h – l – 1 where l is an upper bound for zeros of the corresponding polynomials (and which can be effectively estimated in terms of their heights)

scholarly journals On a Quasi-Linear Equation

1956 ◽  
Vol 8 ◽  
pp. 198-202 ◽  
Author(s):  
Richard Bellman
Keyword(s):  
Linear Equation ◽  
Limit Theorems ◽  
Recurrence Relations ◽  
Linear Recurrence ◽  
Initial Values ◽  
Non Linear ◽  
Image Position

1. Introduction. The purpose of this note is to establish some limit theorems for the non-linear recurrence relations1.1, i= 1,2, …, N; n ≥ 0under certain assumptions concerning the initial values ci = xi(0), and the coefficient matrices A (q) = (aij(q)).

scholarly journals Supplement to a Note on Recurrent Sequences

1933 ◽  
Vol 3 (3) ◽  
pp. 220-222
Author(s):  
C. E. Walsh
Keyword(s):  
Positive Integer ◽  
Recurrence Relation ◽  
Initial Values ◽  
Recurrent Sequences ◽  
Image Position

A lemma used in the above paper can be extended to cover the case of a sequence Sn determined by a recurrence relation of the formr being any positive integer.For convenience denote everywhere. Let us suppose that an inequality of the formholds for p = 2, 3, … ., n — 1. The constant K can be determined from the initial values of Sn, when kn and ln have been found.

scholarly journals A recurrence relation generalizing those of Apéry

1984 ◽  
Vol 36 (2) ◽  
pp. 267-278 ◽  
Author(s):  
Richard Askey ◽  
J. A. Wilson
Keyword(s):  
Recurrence Relation ◽  
Recurrence Relations ◽  
Image Position ◽  
Three Term Recurrence Relation

AbstractA three term recurrence relation is found forwhen a + d = b + c. This includes the recurrence relations of Apéry associated with ζ(3), ζ(2) and log 2 as special or limiting cases.

scholarly journals Simultaneous Linear Recurrence Relations with Variable Coefficients

1958 ◽  
Vol 9 (4) ◽  
pp. 183-206
Author(s):  
H. D. Ursell
Keyword(s):  
Recurrence Relation ◽  
Recurrence Relations ◽  
Standard Form ◽  
Variable Coefficients ◽  
Linear Recurrence ◽  
Integer Variable ◽  
Self Consistent ◽  
Fixed Order ◽  
Image Position ◽  
Linear Recurrence Relation

Our subject is a set of equationswhere the uj(n)(j = 1, 2, …, k) are k “unknown” functions of the integer variable n, the zi(n) (i = 1, 2, … h) are h “known” functions of n, and the Aij(n) are hk “known” operatorswhich are polynomials in E, each of fixed order pij but with coefficients which may vary with n. E is the usual operator defined byOur first task is to determine whether the equations (1) are self-consistent. Secondly, if they are self-consistent, we ask what follows from them for a given subset of the unknowns, e.g. for (uj+1, …, uk) in other words we wish to eliminate (u1, …, uj). In particular we wish to eliminate all the variables but one, say uk. We shall in fact find that either uk is arbitrary or else that it has only to satisfy a single linear recurrence relation : and the order of that relation is of interest to us. Thirdly, we ask that reduction to standard form is possible, with or without a transformation of the unknowns themselves.

scholarly journals On the Sum of Powers of Two k-Fibonacci Numbers which Belongs to the Sequence of k-Lucas Numbers

2016 ◽  
Vol 67 (1) ◽  
pp. 41-46
Author(s):  
Pavel Trojovský
Keyword(s):  
Recurrence Relation ◽  
Initial Conditions ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Lucas Numbers ◽  
Initial Values ◽  
Lucas Sequence ◽  
Positive Integers

Abstract Let k ≥ 1 and denote (Fk,n)n≥0, the k-Fibonacci sequence whose terms satisfy the recurrence relation Fk,n = kFk,n−1 +Fk,n−2, with initial conditions Fk,0 = 0 and Fk,1 = 1. In the same way, the k-Lucas sequence (Lk,n)n≥0 is defined by satisfying the same recurrence relation with initial values Lk,0 = 2 and Lk,1 = k. These sequences were introduced by Falcon and Plaza, who showed many of their properties, too. In particular, they proved that Fk,n+1 + Fk,n−1 = Lk,n, for all k ≥ 1 and n ≥ 0. In this paper, we shall prove that if k ≥ 1 and $F_{k,n + 1}^s + F_{k,n - 1}^s \in \left( {L_{k,m} } \right)_{m \ge 1} $ for infinitely many positive integers n, then s =1.

scholarly journals ELEMENTARY OSCILLATION CRITERIA FOR A THREE TERM RECURRENCE RELATION WITH OSCILLATORY COEFFICIENT SEQUENCE

1998 ◽  
Vol 29 (3) ◽  
pp. 227-232
Author(s):  
GUANG ZHANG ◽  
SUI-SUN CHENG
Keyword(s):  
Recurrence Relation ◽  
Recurrence Relations ◽  
Qualitative Properties ◽  
Oscillation Criteria ◽  
Three Term Recurrence Relation

Qualitative properties of recurrence relations with coefficients taking on both positive and negative values are difficult to obtain since mathematical tools are scarce. In this note we start from scratch and obtain a number of oscillation criteria for one such relation : $x_{n+1}-x_n+p_nx_{n-r}\le 0$.

scholarly journals On the quotient moment of lower generalized order statistics and characterization

2015 ◽  
Vol 11 (1) ◽  
pp. 73-89
Author(s):  
Devendra Kumar
Keyword(s):  
Order Statistics ◽  
Recurrence Relation ◽  
Conditional Expectation ◽  
General Class ◽  
Recurrence Relations ◽  
Generalized Order Statistics ◽  
Lower Records ◽  
Moments Of Order Statistics ◽  
Generalized Order

Abstract In this paper we consider general class of distribution. Recurrence relations satisfied by the quotient moments and conditional quotient moments of lower generalized order statistics for a general class of distribution are derived. Further the results are deduced for quotient moments of order statistics and lower records and characterization of this distribution by considering the recurrence relation of conditional expectation for general class of distribution satisfied by the quotient moment of the lower generalized order statistics.

scholarly journals Distinct partitions and overpartitions

2021 ◽  
Vol 38 (1) ◽  
pp. 149-158
Author(s):  
MIRCEA MERCA ◽  
Keyword(s):  
Partition Function ◽  
Recurrence Relation ◽  
Recurrence Relations ◽  
Convergent Series ◽  
The Other ◽  
Large Integer ◽  
Linear Recurrence ◽  
Fast Computation ◽  
Real Numbers ◽  
Very High

In 1963, Peter Hagis, Jr. provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the partition function $Q(n)$ which counts partitions of $n$ into distinct parts. Computing $Q(n)$ by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we investigate new connections between partitions into distinct parts and overpartitions and obtain a surprising recurrence relation for the number of partitions of $n$ into distinct parts. By particularization of this relation, we derive two different linear recurrence relations for the partition function $Q(n)$. One of them involves the thrice square numbers and the other involves the generalized octagonal numbers. The recurrence relation involving the thrice square numbers provide a simple and fast computation of the value of $Q(n)$. This method uses only (large) integer arithmetic and it is simpler to program. Infinite families of linear inequalities involving partitions into distinct parts and overpartitions are introduced in this context.

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