The Generating Function of a Bi-Periodic Leonardo Sequence

. In “A note on bi-periodic Leonardo sequence”, the generating function for a certain bi-periodic Leonardo sequence is claimed. In this note, based on the literature, we establish the correct identity. Possible periodic extensions for the Leonardo sequence are discussed, opening new avenues for results in the area.


The Leonardo sequence and a bi-periodic extension
The Leonardo sequence (Le n ) is defined by the inhomogeneous recurrence relation Le n = Le n−1 + Le n−2 + 1 for n 2, with initial conditions Le 0 = Le 1 = 1 .
In the recent work [6], Catarino and Spreafico proposed a bi-periodic extension of Leonardo sequence, say (Le (a,b)   n ), defined by the recurrence relations Le (a,b) In [6,Theorem 3], the authors claim that the generating function for the bi-periodic Leonardo sequence (Le In this note we show that this statement is not accurate.We provide the correct generating function and corresponding proof based on existing literature.This will be done in the next section where the main tools will also be provided.Our approach is of matricial nature.In the last section, we propose a new bi-periodic Leonardo sequence and possible extensions.

Recurrence relations, Hessenberg matrices, and generating functions
Let us consider the sequence (a n ) defined by the homogeneous recurrence relation for n > r, with given initial conditions It is well-known that (a n ) can be written explicitly as the determinant of a Hessenberg matrix, namely, For details and general applications, the reader is referred to [14,18,22].A general result can be found, for example, in [21,Theorem 4.20].Furthermore, if we replace the −1's of the subdiagonal of the Hessenberg matrix defined in ( 7) by 1's, then a n is the permanent of such matrix (cf.[8]).
Using classical results on Hessenberg matrices it is possible to find in many instances explicit formulas or the generating function for (7).For example, in [11] we have the following result.
Theorem 1 The generating function of the sequence (d n ) defined by the determinants The reader is also invited to read [13] for an earlier look and historical context and [15] for other extensions.Note that we assume k < .
From (7) and taking into account the recurrence relation (4), we obtain for Le Using elementary operations over the first 5 columns of the Hessenberg matrix defined above, the determinant equals From Theorem 1, we finally get A straightforward factorization of the denominator lead us to the following theorem.

Theorem 2
The generating function for the bi-periodic Leonardo sequence ) .

An extension for the bi-periodic Leonardo sequence
The main motivation for introducing the bi-periodic Leonardo sequence (Le (a,b)

n
) is the bi-periodic Horadam sequence (h n ) defined by the homogeneous recurrence relations for n 2, with initial conditions h 0 = a and h 1 = ab + 1.This sequence can be deduced by the determinantal identity for n 1, and can be determined in terms of Chebyshev polynomials of the second kind.These matrices are called bi-periodic, and the study of their determinants, called continuants, dates back to the 1940's (or, perhaps, even earlier) in the context of physics and chemistry with the article [20].See also [7].
Consequently, we proposed a first new bi-periodic Leonardo sequence, say generically ( Le Another possibility which might be of interest is when the recurrence relations satisfy Le Certainly, by changing the initial conditions, we will obtain new sequences that will deserve study.We chose these because they seemed to be the best suited to the proposal in [6].We can also aim for similar relations for the tri-periodic case as in [6].We leave open to the interested reader a more in-depth study of these extensions.