Revista Tocantinense de Matematica
Vol. 1, n. 1, 2026, p. 1–9
DOI: 10.20873/retmat.uft.v1n1.2026.23181
Received: May 27, 2026
Accepted: June 1, 2026
First-order expansions of coupled Leonardo sequences
Fernando S. de Carvalho1,*
Abstract
In this paper, we investigate the sensitivity of coupled Leonardo sequences under perturba-
tions of the recurrence coefficients and the coupling term. By introducing a perturbation
parameter, we derive an explicit first-order asymptotic expansion and establish a linear
sensitivity equation governing the leading correction term. Under suitable spectral assump-
tions, we obtain uniform error estimates and show that the perturbed sequence admits
a first-order approximation whose accuracy is controlled by the perturbation magnitude.
These results provide a quantitative description of the stability and local behavior of
coupled Leonardo sequences and contribute to the perturbation theory of linear recursive
systems.
Keywords: Coupled Leonardo sequence; perturbation; stability; spectral radius.
MSC 2020: 39A10; 39A30; 11B39.
Contents
1 Introduction 1
2 Equilibrium and spectral analysis 2
3 Stability with respect to perturbations 4
4 Sensitivity analysis and first-order expansion 6
5 Instability 8
6 Concluding remarks 9
1 Introduction
The Leonardo sequence, listed as
A
001595 in the On-Line Encyclopedia of Integer Sequences
(OEIS) [
1
], is a classical example of a second-order linear recurrence closely related to the
Fibonacci sequence. The Leonardo sequence is defined by
Ln+2 =Ln+1 +Ln+ 1, L0=L1= 1, n ∈N.
1
Universidade Federal do Tocantins (UFT), Campus de Arraias, Tocantins, Brasil. E-mail:
fscarvalho@uft.edu.br. ORCID: 0000-0001-6639-0716.
*Autor correspondente.
ReTMAT 2
The relationship between the Fibonacci sequence
{Fn}n≥0
(
A
000045 in the OEIS [
1
]) and
the Leonardo sequence {Ln}n≥0is given by
Ln= 2(Fn+Fn−1)−1=2Fn+1 −1.
In [
2
], the coupled Leonardo sequence was introduced as a generalization of Leonardo-type
sequences associated with a second-order Horadam recurrence. In its basic form, the sequence
is defined by
L(k,t)
n=kL(k,t)
n−1+tL(k,t)
n−2+Q(k, t), n ≥2,
with initial conditions
L(k,t)
0
=
L(k,t)
1
= 1 and
k, t ∈N
,
kt
= 0. Equation
(1)
preserves the
second-order structure of Horadam-type recurrences while introducing a coupling function
Q(k, t).
The purpose of the present paper is to propose a different type of generalization. Instead
of increasing the order of the recurrence or introducing a non-constant term, we study the
stability of the coupled Leonardo sequence under perturbations of its coefficients and coupling
term. More precisely, we consider
L(ε)
n(k, t)=(k+εf(k, t))L(ε)
n−1+ (t+εg(k, t))L(ε)
n−2+Q(k, t) + εh(k, t)
| {z }
Qε(k,t)
, n ≥2,(1)
where
f, g, h
:
N×N→R
are non-vanishing functions,
ε∈R
is a small perturbation parameter,
and
L(ε)
0=a, L(ε)
1=b,
with a, b ∈N.
The formulation of equation
(1)
has several advantages. In particular, it preserves the
second-order structure of the original sequence and allows one to study how the sequence
varies under small perturbations of the parameters.
The paper is organized as follows. In Section 2we study the equilibrium solution and
the spectral properties associated with the perturbed recurrence. In Section 3we establish
stability estimates with respect to the perturbation parameter. Section 4is devoted to the
sensitivity sequence and to the derivation of the first-order asymptotic expansion. Finally,
Section 5discusses the instability regime when the spectral radius is greater than one.
2 Equilibrium and spectral analysis
To analyze the perturbed recurrence
(1)
, we begin by identifying its equilibrium solution
and reducing the non-homogeneous equation to a homogeneous one. This transformation
allows the dynamics to be expressed in terms of a linear recurrence with perturbed coefficients,
whose behavior is determined by its characteristic roots.
Lemma 2.1. Assume that, 1
−k−t−ε
(
f
(
k, t
) +
g
(
k, t
))
= 0. Then the perturbed recurrence
(1)admits the constant solution
L∗
ε=Qε(k, t)
1−k−t−εf(k, t)+g(k, t).(2)
Proof. Substituting the constant sequence L(ε)
n≡L∗
εinto (1), we obtain
L∗
ε=k+εf(k, t)L∗
ε+t+εg(k, t)L∗
ε+Qε(k, t).
ReTMAT 3
Hence, 1−k−t−ε(f(k, t) + g(k, t))L∗
ε=Qε(k, t),
which proves the expression for L∗
ε.
Note that if we consider
U(ε)
n=L(ε)
n(k, t)−L∗
ε,(3)
we obtain
U(ε)
n=k+εf(k, t)U(ε)
n−1+t+εg(k, t)U(ε)
n−2.(4)
The introduction of the sequence
U(ε)
n
removes the non-homogeneous term from the
recurrence and reduces the problem to a homogeneous linear relation with perturbed coefficients.
This decomposition separates the equilibrium component from the dynamical behavior.
Although the sequence
U(ε)
n
will be used to control the dynamical component of the
recurrence, our main object remains the perturbed coupled Leonardo sequence (1). Since
L(ε)
n=U(ε)
n+L∗
ε,(5)
the stability of
L(ε)
n
follows from the joint control of the homogeneous component
U(ε)
n
and of
the equilibrium L∗
ε.
The homogeneous recurrence associated with U(ε)
nhas characteristic polynomial
pε(x)=x2−(k+εf(k, t))x−(t+εg(k, t)).(6)
Let ρ1(ε)and ρ2(ε)be its roots. We define the spectral radius by
r(ε) = max{|ρ1(ε)|,|ρ2(ε)|}.(7)
Proposition 2.2. The condition r(ε)<1holds if, and only if
|t+εg(k, t)|<1,
1−k−t−εf(k, t)+g(k, t)>0,
and
1+k−t+εf(k, t)−g(k, t)>0.
Proof.
The result follows from the Jury stability criterion for second-order polynomials. More
precisely, the roots of a polynomial of the form
x2−Ax −B
lie inside the unit disk (stability region) if and only if
|B|<1,1−A−B > 0,1 + A−B > 0;
see, for instance, [4]. Applying this criterion with
A=k+εf(k, t), B =t+εg(k, t),
yields the desired conditions.
ReTMAT 4
✁ ✂ ✄ ✂ ✁
✂
✄ ☎ ✆
✄
✄☎ ✆
✂
✝
✞
✟✟ ✠✠ ✡✡ ☛☛ ☞☞ ✌✌ ☞☞ ✠✠ ✍✍ ✎✎✏✏ ✑✑ ☞☞ ✒✒ ✓✓
✔ ✕ ✂ ✖ ✗ ✔ ✕ ✂ ✗
✔ ✕ ✂
Figure 1: Stability region for x2−Ax −B= 0.
The stability region can be visualized in the (
A, B
)-plane, where
A
=
k
+
εf
(
k, t
)and
B
=
t
+
εg
(
k, t
). In this setting, the condition
r
(
ε
)
<
1corresponds to a triangular region, as
illustrated in Figure 1.
Proposition 2.3 shows that, under the condition
r
(
ε
)
<
1, the perturbed sequence
(1)
converges to its equilibrium solution (2).
Proposition 2.3. Assume that the spectral radius
r
(
ε
)associated with the homogeneous
recurrence (5)satisfies r(ε)<1. Then,
L(ε)
n−→ L∗
εas n→ ∞.
Proof. From the Equation (3), we have
L(ε)
n−L∗
ε=U(ε)
n.
Since r(ε)<1, we have
U(ε)
n→0as n→ ∞.
It follows that
L(ε)
n→L∗
ε,
which completes the proof.
3 Stability with respect to perturbations
In this section, we study the dependence of the perturbed recurrence on the parameter
ε
. In particular, we establish estimates describing the stability of the sequence under small
perturbations of the coefficients and of the coupling term.
Proposition 3.1. For each fixed n≥0, the mapping
ε7→ L(ε)
n
is a polynomial function of εof degree at most n−1.
ReTMAT 5
Proof.
We proceed by induction on
n
. For
n
= 0
,
1, by the definition of the initial conditions
in (1), the terms
L(ε)
0=a, L(ε)
1=b, a, b ∈N,
are constant functions of ε, hence polynomials.
Assume that L(ε)
n−1and L(ε)
n−2are polynomials in ε. Then, from the recurrence,
L(ε)
n=k+εf(k, t)L(ε)
n−1+t+εg(k, t)L(ε)
n−2+Q(k, t) + εh(k, t),
it follows that
L(ε)
n
is a linear combination of polynomials, and hence also a polynomial. The
result follows by induction.
Since the initial data do not depend on
ε
, the first-order variation starts only through
the recurrence itself. This leads to an auxiliary nonhomogeneous recurrence driven by the
unperturbed sequence. Corollary 3.2 follows immediately from the polynomial dependence.
Corollary 3.2. For each fixed n≥0, one has
lim
ε→0L(ε)
n=L(0)
n.
Proof.
The result follows immediately from Proposition 3.1, since for each fixed
n
, the mapping
ε7→ L(ε)
nis continuous.
Theorem 3.3 provides a uniform control of the perturbation with respect to both the
parameter εand the index n.
Theorem 3.3. Assume that
r
(0)
<
1. Then there exist constants
ε0>
0and
C >
0such that,
for every |ε|< ε0,
sup
n≥0
|L(ε)
n−L(0)
n|≤C|ε|.
Proof. By equation (5), we have
L(ε)
n=U(ε)
n+L∗
ε, L(0)
n=U(0)
n+L∗
0.
Hence,
|L(ε)
n−L(0)
n| ≤ |U(ε)
n−U(0)
n|+|L∗
ε−L∗
0|.
Since
r
(0)
<
1, by continuity of the roots there exist
ε0>
0and 0
< θ <
1such that
r
(
ε
)
≤θ
for all
|ε|< ε0
. Thus the homogeneous solutions satisfy a uniform exponential bound:
there exists C1>0such that
|U(ε)
n|≤C1θn,|U(0)
n|≤C1θn,∀n≥0,|ε|< ε0.
Set
Vn=U(ε)
n−U(0)
n.
Using equation (4), we obtain
Vn= (k+εf(k, t))Vn−1+ (t+εg(k, t))Vn−2+εf (k, t)U(0)
n−1+εg(k, t)U(0)
n−2.
Since
r
(
ε
)
≤θ <
1, the homogeneous recurrence associated with
Vn
is uniformly exponentially
stable. Moreover, the forcing term is of order
O
(
εθn
). Standard estimates for stable linear
recurrences then imply the existence of a constant C2>0such that
sup
n≥0
|U(ε)
n−U(0)
n|≤C2|ε|.
ReTMAT 6
Moreover, from the explicit formula for L∗
εand the condition 1−k−t= 0, the map
ε7→ L∗
ε
is differentiable in a neighborhood of 0. Hence, there exists C3>0such that
|L∗
ε−L∗
0| ≤ C3|ε|.
Combining the previous estimates, we obtain
sup
n≥0
|L(ε)
n−L(0)
n| ≤ (C2+C3)|ε|.
This completes the proof.
4 Sensitivity analysis and first-order expansion
The study of sensitivity with respect to parameters plays an important role in the analysis
of discrete dynamical systems. In the context of linear recurrences, it provides a quantitative
description of how solutions vary under small perturbations of the coefficients and coupling
terms, as considered in equation
(1)
. This type of analysis is closely related to perturbation
theory and is used to assess the stability and robustness of discrete models.
The main purpose of this section is to derive the first-order expansion of the coupled
Leonardo sequence under parameter perturbations and to identify the auxiliary recurrence
governing its sensitivity (see [3,4]).
Definition 4.1. The sensitivity sequence {Zn}n≥0is defined by
Zn=d
dεL(ε)
n
ε=0
, n ≥0.
The following result provides a recurrence satisfied by the sensitivity sequence Zn.
Proposition 4.2. The sensitivity sequence {Zn}n≥0satisfies
Z0= 0, Z1= 0,
and, for n≥2,
Zn=kZn−1+tZn−2+f(k, t)L(0)
n−1+g(k, t)L(0)
n−2+h(k, t).(8)
Proof.
Differentiating the perturbed recurrence given in equation
(1)
with respect to
ε
, we
obtain
d
dεL(ε)
n=f(k, t)L(ε)
n−1+ (k+εf(k, t)) d
dεL(ε)
n−1+g(k, t)L(ε)
n−2+ (t+εg(k, t)) d
dεL(ε)
n−2+h(k, t).
Evaluating at ε= 0, and using the definition 4.1, yields
Zn=kZn−1+tZn−2+f(k, t)L(0)
n−1+g(k, t)L(0)
n−2+h(k, t).
Since the initial conditions L(ε)
0=aand L(ε)
1=bdo not depend on ε, it follows that
Z0=d
dεL(ε)
0
ε=0
=d
dεa= 0, Z1=d
dεL(ε)
1
ε=0
=d
dεb= 0.
ReTMAT 7
The recurrence satisfied by
Zn
shows that the sensitivity sequence is driven by the unper-
turbed dynamics
L(0)
n
. In particular, the first-order response inherits the spectral structure of
the original recurrence while incorporating the perturbed terms associated with f,g, and h.
Theorem 4.3 shows that the sequence
Zn
provides the first-order term in the asymptotic
expansion of L(ε)
n.
Theorem 4.3. For each fixed n≥0, one has
L(ε)
n=L(0)
n+εZn+O(ε2), ε →0.(9)
Proof. Fix n≥0. By Proposition 3.1, the mapping
ε7→ L(ε)
n
is a polynomial function of
ε
. In particular, it is twice continuously differentiable in a
neighborhood of
ε
= 0. By Taylor’s formula with remainder, there exists
ξε
between 0and
ε
such that
L(ε)
n=L(0)
n+εd
dεL(ε)
n
ε=0
+ε2
2
d2
dε2L(ε)
n
ε=ξε
.
By the definition 4.1,
Zn=d
dεL(ε)
n
ε=0
.
Hence,
L(ε)
n=L(0)
n+εZn+ε2
2
d2
dε2L(ε)
n
ε=ξε
.
Since L(ε)
nis a polynomial in ε, its second derivative is continuous and therefore bounded
in some interval |ε| ≤ ε0. Thus, there exists a constant Cn>0such that
d2
dε2L(ε)
n
ε=ξε
≤Cn,
for all sufficiently small ε. Consequently,
L(ε)
n−L(0)
n−εZn
≤Cn
2ε2,
which proves that
L(ε)
n=L(0)
n+εZn+O(ε2), ε →0.
The representation of the recurrence
(9)
separates the dependence on
ε
from the intrinsic
dynamics of the sequence. In particular,
L(0)
n
describes the unperturbed behavior, while
Zn
captures the first-order response to perturbations.
Remark 4.4.The expansion obtained in Theorem 4.3 is pointwise with respect to the index
n
. In general, the constant involved in the
O
(
ε2
)term may depend on
n
. Uniform first-order
expansions require additional stability assumptions, such as a spectral radius strictly smaller
than one.
ReTMAT 8
5 Instability
In Sections 3and 4, it was shown that when the spectral radius is strictly less than
one, the effect of perturbations remains controlled. In the present section, we analyze the
complementary case in which the spectral radius is greater than one.
Proposition 5.1. Assume that
r
(0)
>
1. Suppose that the unperturbed characteristic polyno-
mial (6)has a simple dominant root ρ0, that is,
|ρ0|>max{1,|σ0|},
where
σ0
denotes the other characteristic root. Assume also that the coefficient of the dominant
mode in the homogeneous component of L(0)
nis nonzero and that
ρ′(0) = 0,
where
ρ
(
ε
)denotes the continuation of the dominant root for
ε
near 0. Then the convergence
L(ε)
n→L(0)
n
as ε→0is not uniform in n.
Proof.
Let
U(ε)
n
=
L(ε)
n−L∗
ε
(Equation
(3)
). By Equation
(4)
,
U(ε)
n
satisfies the homogeneous
recurrence
U(ε)
n= (k+εf(k, t))U(ε)
n−1+ (t+εg(k, t))U(ε)
n−2.
Let ρ(ε)and σ(ε)be the roots of the perturbed characteristic polynomial, with
ρ(0) = ρ0, σ(0) = σ0.
Since
ρ0
is a simple dominant root, for
|ε|
sufficiently small the roots depend smoothly on
ε
,
and ρ(ε)remains dominant. Thus the homogeneous component can be written as
U(ε)
n=C(ε)ρ(ε)n+D(ε)σ(ε)n,
where C(ε)and D(ε)depend on the initial conditions. By hypothesis,
C(0) = 0.
Since ρ′(0) = 0, for sufficiently small nonzero εone has
ρ(ε)=ρ(0).
Moreover, since |ρ0|>1, we still have
|ρ(ε)|>1
for εsmall.
Now,
L(ε)
n−L(0)
n=U(ε)
n−U(0)
n+L∗
ε−L∗
0.
The second term is independent of
n
, and therefore cannot compensate the growth of the
homogeneous part.
Using the representations of U(ε)
nand U(0)
n, we obtain
U(ε)
n−U(0)
n=C(ε)ρ(ε)n−C(0)ρn
0+D(ε)σ(ε)n−D(0)σn
0.
ReTMAT 9
Since ρ(ε)and ρ0are distinct dominant roots and |ρ0|>1, the dominant part
C(ε)ρ(ε)n−C(0)ρn
0
does not remain bounded as
n→ ∞
, except in degenerate cases excluded by the assumptions.
Hence
sup
n≥0
|L(ε)
n−L(0)
n|= +∞
for sufficiently small nonzero ε. In particular,
sup
n≥0
|L(ε)
n−L(0)
n| −→ 0,as ε→0.
Therefore, the convergence is not uniform in n.
6 Concluding remarks
In this paper, we analyzed the stability of perturbed coupled Leonardo sequences under
simultaneous perturbations of the recurrence coefficients and of the coupling term. The
equilibrium structure of the recurrence was characterized through a spectral approach, leading
to stability and instability criteria in terms of the spectral radius.
We also introduced the sensitivity sequence associated with the perturbation parameter
and derived the first-order asymptotic expansion
L(ε)
n=L(0)
n+εZn+O(ε2),
which quantitatively describes the response of the sequence to small perturbations.
These results provide a framework for the analysis of coupled Leonardo-type recurrences
under small perturbations and may be adapted to other families of linear recurrences with
nonhomogeneous structure.
Acknowledgements
The author thanks PROPESQ/UFT for the institutional support.
References
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N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences; The OEIS Foundation
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[2]
E. G. Mesquita, F. Alves, F. Carvalho, and E. Costa, Some properties of the coupled
Leonardo sequence, Hacettepe Journal of Mathematics and Statistics, 2026.
[3] S. Elaydi, An Introduction to Difference Equations, 3rd ed., Springer, 2005.
[4]
W. G. Kelley and A. C. Peterson, Difference Equations: An Introduction with Applications,
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