ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P1.08
https://doi.org/10.26493/1855-3974.2473.f2e
(Also available at http://amc-journal.eu)
New strong divisibility sequences*
Zhibin Du
School of Software, South China Normal University, Foshan, Guangdong 528225, China
Darko Dimitrov
Faculty of Information Studies, 8000 Novo mesto, Slovenia
Carlos M. da Fonseca †
Kuwait College of Science and Technology,
Doha District, Block 4, P.O. Box 27235, Safat 13133, Kuwait, and
Chair of Computational Mathematics, University of Deusto,
48007 Bilbao, Basque Country, Spain
Received 28 October 2020, accepted 26 June 2021, published online 5 April 2022
Abstract
We provide new families of divisibility and strong divisibility sequences based on some
factorization properties of Chebyshev polynomials.
Keywords: Strong divisibility sequences, Chebyshev polynomials, tridiagonal matrices, determinant.
Math. Subj. Class. (2020): 11B37, 11B39, 11B83, 15B05, 33C45
1 Introduction
A sequence of any integer numbers {an} is said to be a divisibility sequence if
am | an, whenever m | n,
and is called a strong divisibility sequence if
gcd(am, an) = agcd(m,n).
*This work was supported by the National Natural Science Foundation of China (Grant No. 11701505).
†Corresponding author.
E-mail addresses: zhibindu@126.com (Zhibin Du), darko.dimitrov11@gmail.com (Darko Dimitrov),
c.dafonseca@kcst.edu.kw (Carlos M. da Fonseca)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Ars Math. Contemp. 22 (2022) #P1.08
The strong divisibility sequences and its weaker version have been studied for more
than one century. Actually, the Fibonacci numbers
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, . . .
are perhaps the best known non-trivial strong divisibility sequence. For earlier questions,
open problems, and general characterizations, the reader is referred to [4, 10, 11, 12, 21,
22].
As a particular case of the general conditional recurrence sequences defined in [16],
recently it was proposed in [20] the study of the conditional recurrence sequences {fn}
satisfying the recurrence relations of integers
fn =
{
a1fn−1 + b2fn−2, if n is odd,
a2fn−1 + b1fn−2, if n is even.
for n ⩾ 2, with f0 = 1 and f1 = a1, aiming to generate new strong divisibility sequences.
Indeed, the authors were able to obtain sufficient conditions for which certain subsequences
of {fn} are strong divisible.
Theorem 1.1 ([20]). Let f̃n = f2n−1. If a1 = 1 and gcd(a1a2 + b1 + b2, b1b2) = 1, then
gcd(f̃m, f̃n) = f̃gcd(m,n).
Corollary 1.2 ([20]). Let f̃n = f2n−1. If gcd(a1a2 + b1 + b2, b1b2) = 1, then {f̃n} is a
strong divisibility sequence.
Theorem 1.3 ([20]). Let f̃n = f2n−1. Thus f̃m | f̃n, whenever m | n.
For example, setting a1 = 3, a2 = 1 = b1, and b2 = 2, we get
n 1 2 3 4 5 6 7 8 9
fn 3 4 18 22 102 124 576 700 3252
This means that the first terms of the subsequence of odd indices of {fn} are
n 1 2 3 4 5 6
f̃n 3 18 102 576 3252 18360
While {f̃n} is a divisibility sequence, it is clear that is not strong.
Another interesting result obtained in [20] is the following:
Theorem 1.4. Let f̃1 = 1 and f̃n = fn−1, for n > 1. If a1 = 1, b1 = b2, and
gcd(a2, b1) = 1, then {f̃n} is a strong divisibility sequence.
For the weaker divisibility, the following general result was obtained:
Corollary 1.5. Under the conditions of Theorem 1.4, {f̃n} is a divisibility sequence.
Z. Du et al.: New strong divisibility sequences 3
Our aim here is to extend the above results to a more general setting, namely for the
sequences of integers defined by the recurrence relations
fn =

a1fn−1 + bkfn−2, if n ≡ 1 (mod k),
a2fn−1 + b1fn−2, if n ≡ 2 (mod k),
a3fn−1 + b2fn−2, if n ≡ 3 (mod k),
· · · · · ·
ak−1fn−1 + bk−2fn−2, if n ≡ k − 1 (mod k),
akfn−1 + bk−1fn−2, if n ≡ 0 (mod k),
(1.1)
for n ⩾ 2, with f0 = 1 and f1 = a1. The previous results will be recovered by making
k = 2. Consequently, we answer to the open problem proposed in [20].
In this paper, we will relate (1.1) with the so-called periodic continuants [6, 18] (for
recent applications, the reader is referred to [1, 2, 3]). This relation is established by using
Chebyshev polynomials of the second kind. Then, from {fn} we can, under certain con-
ditions, generate new strong divisibility sequences. At the same time, we can recover the
connection between the sequences defined by recurrence relations with two terms and the
determinants of tridiagonal matrices. This is effectively in the spirit of some ideas we can
find in [15], proposed by Édouard Lucas back to 1878.
2 The determinant of a tridiagonal k-Toeplitz matrix
The matrices of the form
An =

a1 b1
c1
. . . . . .
. . . ak bk
ck a1 b1
c1
. . . . . .
. . . ak bk
ck a1 b1
c1
. . . . . .
. . .

n×n
,
i.e., tridiagonal matrices An = (aij) with entries satisfying
ai+k,j+k = aij , for i, j = 1, 2, . . . , n− k,
are known as tridiagonal k-Toeplitz. The determinant of such matrix is known as a periodic
continuant [18].
For k = 1, we get a tridiagonal Toeplitz matrix and its determinant was known in [18]
as a continuant. The characteristic polynomial of such a matrix was found by V. Lovass-
Nagy and P. Rózsa [13, 14], in 1963. Notwithstanding, the particular case when k = 2 and
the two subdiagonals are constant equal to 1, had been considered in 1947 in D. E. Ruther-
ford’s seminal paper [19], followed soon after by J. F. Elliott with his Master’s thesis [5,
4 Ars Math. Contemp. 22 (2022) #P1.08
Section IV.4]. In 1966, Rózsa held a seminar at the University of Hamburg on tridiagonal
k-Toeplitz matrices motivated mainly by problems of lattice dynamics, of ladder networks,
and of structural analysis. In that year, L. Elsner and R. M. Redheffer [6] studied An for
special cases of k and, two years later, P. Rózsa in [18] originally proved a general formula
for the determinant of An. Independently, the spectrum of a tridiagonal 2-Toeplitz matrix
was also studied by M. J. C. Gover in 1994 [9]. In [7], it is considered the case when k = 3
and, later on, the characteristic polynomial of An was stated, for any k, when analyzing the
invertibility conditions for An based on orthogonal polynomials theory (cf. [8]).
We recall now Rózsa’s solution. Let ∆i1,i2,...,ip be the principal minor of An indexed
by the rows and columns i1, i2, . . . , ip. The determinant of An is given in [18] as
detAn = (
√
b1c1 · · · bkck)q
(
∆1...,rUq(x)+
√
bkckb1c1 · · · brcr√
br+1cr+1 · · · bk−1ck−1
∆r+2,...,k−1 Uq−1(x)
)
with n = qk + r and
x =
∆1,...,k − bkck∆2,...,k−1
2
√
b1c1 · · · bkck
,
assuming that ∆1,...,r = 1 and ∆2,...,r = 0, for r = 0, and with {Un(x)}n⩾0 standing for
the Chebyshev polynomials of the second kind. These polynomials satisfy the three-term
recurrence relation
Un+1(x) = 2xUn(x)− Un−1(x), for all n = 1, 2, . . . , (2.1)
with initial conditions U0(x) = 1 and U1(x) = 2x. We recall that the main explicit formula
for the Chebyshev polynomials of the second kind could be
Un(x) =
sin(n+ 1)θ
sin θ
, with x = cos θ (0 ⩽ θ < π), (2.2)
for all n = 0, 1, 2 . . . . While (2.2) is more common to find in the orthogonal polynomials
theory, there are other explicit representations and relations for Un(x). Among them, the
most frequent are
Un(x) =
(
x+
√
x2 − 1
)n+1 − (x−√x2 − 1)n+1
2
√
x2 − 1
,
an immediate consequence of de Moivre’s formula, and
Un(x) =
⌊n2 ⌋∑
k=0
(−1)k
(
n− k
k
)
(2x)n−2k.
Taking into account the definition of An, we can redefine (1.1) in terms of the determi-
Z. Du et al.: New strong divisibility sequences 5
nant of An, namely,
fn =
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
a1 b1
−1
. . . . . .
. . . ak bk
−1 a1 b1
−1
. . . . . .
. . . ak bk
−1 a1 b1
−1
. . . . . .
. . .
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
n×n
. (2.3)
That means that the determinant (2.3) is
fn = (i
k
√
b1 · · · bk)q
(
∆1,...,r Uq(x) +
ir+1
√
bkb1 · · · br
ik−r−1
√
br+1 · · · bk−1
∆r+2,...,k−1 Uq−1(x)
)
,
where
x =
∆1,...,k + bk∆2,...,k−1
ik2
√
b1 · · · bk
.
In particular, if r = k − 1, then
fn = (i
k
√
b1 · · · bk)q∆1,...,k−1 Uq(x), (2.4)
which we will explore to generate new strong divisibility sequences in the next sections.
Before that, we recall a general result relating distinct minors, which can be found for
example in [18].
Lemma 2.1. For any positive integer n and i < j,
∆1,...,j−1∆i+1,...,n − (−1)j−ibi · · · bj−1∆1,...,i−1∆j+1,...,n = ∆1,...,n∆i+1,...,j−1.
In fact, Lemma 2.2 in [20] is a particular case of Lemma 2.1.
3 New divisibility sequences
In [20], the authors asked for conditional (strong) divisibility sequences for r > 2, i.e.,
satisfying (1.1). We start with the weaker condition.
Let us recall several factorization properties for Chebyshev polynomials disclosed
in [17].
Theorem 3.1 ([17]). Let m ⩾ n be two positive integers. Then Um(x) is a multiple of
Un(x) if and only if m = (ℓ+ 1)n+ ℓ, for some nonnegative integer ℓ. More precisely, if
ℓ is even, then
Um(x) = Un(x)
2 ℓ2∑
k=0
Tm−(2k+1)n−2k(x)− 1
 ,
6 Ars Math. Contemp. 22 (2022) #P1.08
and if ℓ is odd, then
Um(x) = 2Un(x)
ℓ−1
2∑
k=0
Tm−(2k+1)n−2k(x).
In Theorem 3.1, {Tn(x)}n⩾0 stands for the Chebyshev polynomial of the first kind.
These polynomials satisfy the same recurrence (2.1), here with initial conditions
T0(x) = 1 and T1(x) = x. An explicit formula for such polynomials is Tn(x) = cosnθ,
with x = cos θ.
The next two results, naturally connected to those in Section 1, can be found in [17].
Theorem 3.2. Let m and n be two nonnegative integers and d = gcd(m,n). Then
gcd(Um−1(x), Un−1(x)) = Ud−1(x).
Corollary 3.3. If m and n are coprime, then gcd(Um−1(x), Un−1(x)) = 1.
The general sequences that we consider are
fn = (±
√
b)n−1Un−1
(
a
±2
√
b
)
,
where a, b are nonzero integers (possibly with b < 0), for n ⩾ 1. In particular, f0 = 0,
f1 = 1 and f2 = a.
It is worth mentioning that the symbol ± can be ignored, that is to say:
fn = (±
√
b)n−1Un−1
(
a
±2
√
b
)
= (
√
b)n−1Un−1
(
a
2
√
b
)
, (3.1)
since the Chebyshev polynomials of the second kind Un(x) have the same parities as n.
We may now state our first main result.
Theorem 3.4. For any integers a and b, {fn} as defined in (3.1) is a divisibility sequence.
Proof. Assume that n | m, say m = sn, where s ⩾ 1. For simplicity, set x = 1
2
√
b
. So
fn =
Un−1(ax)
(2x)n−1
and fm =
Usn−1(ax)
(2x)sn−1
,
which implies that
fm
fn
=
Usn−1(ax)
(2x)(s−1)nUn−1(ax)
.
Set ℓ = s− 1, we have sn− 1 = (ℓ+ 1)(n− 1) + ℓ. From Theorem 3.1, Usn−1(x) is
a multiple of Un−1(x). More precisely, when s is even,
Usn−1(x) = 2Un−1(x)
s−2
2∑
t=0
T(s−2t−1)n(x),
Z. Du et al.: New strong divisibility sequences 7
and when s is odd,
Usn−1(x) = Un−1(x)
2 s−12∑
t=0
T(s−2t−1)n(x)− 1
 .
Therefore
fm
fn
=
Usn−1(ax)
(2x)(s−1)nUn−1(ax)
=
2
(2x)(s−1)n
s−2
2∑
t=0
T(s−2t−1)n(ax)
when s is even, and
fm
fn
=
Usn−1(ax)
(2x)(s−1)nUn−1(ax)
=
2
(2x)(s−1)n
s−1
2∑
t=0
T(s−2t−1)n(ax)−
1
(2x)(s−1)n
when s is odd.
We will prove Usn−1(ax)
(2x)(s−1)nUn−1(ax)
is an integer whether s is even or odd, by involving
with the following two claims.
Claim 1. 2T(s−2t−1)n
(
a
2
)
is an integer, for any 0 ⩽ t ⩽ ⌊ s−12 ⌋.
This claim follows immediately from the recurrence relation about Tn(x) as shown
in (2.1).
Claim 2. (
√
b)(s−1)nT(s−2t−1)n
(
1√
b
)
is an integer, for any 0 ⩽ t ⩽ ⌊ s−12 ⌋.
Observe that among all the terms in T(s−2t−1)n
(
1√
b
)
, the maximum degree of denom-
inator is (
√
b)(s−1)n, which means that all the denominators of T(s−2t−1)n
(
1√
b
)
would be
canceled by (
√
b)(s−1)n. It leads to this claim.
Combining the above claims, it leads to
2
(2x)(s−1)n
⌊ s−12 ⌋∑
t=0
T(s−2t−1)n(ax) = 2(
√
b)(s−1)n
⌊ s−12 ⌋∑
t=0
T(s−2t−1)n
(
a
2
√
b
)
is an integer. When s is even, fn | fm follows now. When s is odd, together with the fact
that 1
(2x)(s−1)n
= (
√
b)(s−1)n is an integer, fn | fm also holds.
4 Strong divisibility sequences
The sequence {fn} defined in (3.1) can have negative terms. Therefore, in our strongly
divisibility definition, we are assuming that gcd(am, an) = |agcd(m,n)|. Since we are
interested in positive conditional recurrence sequences (1.1), all the terms of {fn} will be
considered as positive or, equivalently, a > 0 and a2 − 4b ⩾ 0. Notice that the zeros
of the Chebyshev polynomials of the second kind are in the interval (−1, 1) and, from its
definition, limx→+∞ Un(x) = +∞.
8 Ars Math. Contemp. 22 (2022) #P1.08
In order to provide our characterization to the strong divisibility property of {fn}, let
us state several straightforward relations involving fn, as defined in (3.1). From (2.1), we
have
Un
(
a
2
√
b
)
=
a√
b
Un−1
(
a
2
√
b
)
− Un−2
(
a
2
√
b
)
and
fn = afn−1 − bfn−2. (4.1)
A more general identity can be obtained from (2.1), namely
Us+t (x) = Us (x)Ut (x)− Us−1 (x)Ut−1 (x) ,
and then,
fs+t = fs+1ft − bfsft−1. (4.2)
The next result is an extension of some other results we can find in the literature, as for
example related to the Fibonacci numbers.
Lemma 4.1. If gcd(a, b) = 1, then gcd(fn, fn+1) = 1 for any n ⩾ 1.
Proof. We claim that gcd(fn, b) = 1, for any n ⩾ 1, which can be proved by induction.
From f1 = 1 and f2 = a, this claim holds when n = 1, 2. Assume that gcd(fn−1, b) = 1
and gcd(fn−2, b) = 1. Suppose to the contrary that gcd(fn, b) = s, where s > 1. From
(4.1), s | afn−1. Notice that gcd(s, a) = 1, otherwise it is a contradiction to the hypothesis
that gcd(a, b) = 1. So s | fn−1. However, this is another contradiction to the inductive
hypothesis stating gcd(fn−1, b) = 1.
Now we are ready to show that gcd(fn, fn+1) = 1. Again, from f1 = 1 and f2 = a,
we know that gcd(fn, fn+1) = 1 is true when n = 1. Suppose to the contrary that
gcd(fn−2, fn−1) = 1, for some n ⩾ 3, but gcd(fn−1, fn) = t with t > 1. From
(4.1), t | bfn−2. Note that gcd(t, fn−2) = 1, otherwise we get a contradiction with
gcd(fn−2, fn−1) = 1. Thus, t | b means that t is a common divisor of b and fn, a
contradiction to the above claim that gcd(fn, b) = 1.
The proof is now completed.
We are now able to prove the main result of this section.
Theorem 4.2. The sequence {fn} defined in (3.1) is strongly divisible if and only if
gcd(a, b) = 1.
Proof. The necessity part is easy. Assume that {fn} is a strong divisibility sequence.
Suppose to the contrary that gcd(a, b) ̸= 1. From (4.1), we may obtain the first few values:
f1 = 1, f2 = a, f3 = a2 − b, f4 = a3 − 2ab. Clearly, gcd(f3, f4) ̸= 1 = f1 follows from
gcd(a, b) ̸= 1, which is a contradiction to the strong divisibility property of {fn}.
Now we prove the part of sufficiency. Suppose that gcd(a, b) = 1. Set g = gcd(n,m)
and d = gcd(fn, fm). We would like to show that gcd(fn, fm) = |fgcd(n,m)|, i.e., d =
|fg|, which comes from fg | d and d | fg .
On one hand, from g | n and g | m, we get fg | fn and fg | fm, since {fn} is a
divisibility sequence from Theorem 3.4. Thus, fg | d.
On the other hand, we still need to show that d | fg . Since, g = gcd(n,m), we may
assume that there exist positive integers s, k such that sn = g + km. From (4.2), we have
fsn = fg+km = fgfkm+1 − bfg−1fkm.
Z. Du et al.: New strong divisibility sequences 9
From d | fn, and fn | fsn (since {fn} is a divisibility sequence), we get d | fsn. Simi-
larly, we have d | fkm. Therefore, d | fgfkm+1. Notice that gcd(d, fkm+1) = 1, other-
wise, together with d | fkm, it leads to gcd(fkm, fkm+1) ̸= 1, which is a contraction to
Lemma 4.1. Now, it follows that d | fg .
Combining fg | d and d | fg , we obtain d = |fg|, which reveals the strong divisibility
property of {fn}.
5 Examples
In this final section, from the above results, we provide several examples of new (condi-
tional) strong divisibility sequences.
Setting k = 3, r = 2, we have
x =
∆1,...,k + bk∆2,...,k−1
ik2
√
b1 · · · bk
.
In particular, if r = k − 1, then
fn = (−i
√
b1b2b3)
q(a1a2 + b1)Uq
(
a1a2a3 + a3b1 + a1b2 + a2b3
−i2
√
b1b2b3
)
.
So, if we consider the sequence defined by
fn =

fn−1 + 3fn−2, ifn ≡ 1 (mod 3),
2fn−1 + fn−2, ifn ≡ 2 (mod 3),
4fn−1 + 2fn−2, if n ≡ 0 (mod 3),
we have
fn = (−i
√
6)q 3Uq
(
20
−i2
√
6
)
.
Now set
gq+1 = (−i
√
6)q Uq
(
20
−i2
√
6
)
,
for q ⩾ 0. The first terms are:
n gn
1 1
2 20
3 406
4 8240
5 167236
6 3394160
7 68886616
8 1398097280
9 28375265296
10 575893889600
Now, we can check, for example, that g3 | g6 or g5 | g10. However,
gcd(g8, g10) = 320.
10 Ars Math. Contemp. 22 (2022) #P1.08
Instead, we take the recurrence relation
fn =

2fn−1 + 3fn−2, if n ≡ 1 (mod 3),
fn−1 + fn−2, if n ≡ 2 (mod 3),
4fn−1 + 2fn−2, if n ≡ 0 (mod 3).
Setting
gq+1 = (−i
√
6)q Uq
(
19
−i2
√
6
)
,
for q ⩾ 0, the first terms are:
n gn
1 1
2 19
3 367
4 7087
5 136855
6 2642767
7 51033703
8 985496959
9 19030644439
10 367495226095
Now we can check, for example, that g4 | g8 or g5 | g10. Moreover,
gcd(g8, g10) = g2 or gcd(g6, g9) = g3,
and, of course,
gcd(g4, g9) = g1.
Let us consider now two more elaborated examples, for k = 4. We start with the
following one
fn =

2fn−1 + 4fn−2, if n ≡ 1 (mod 4),
fn−1 + 3fn−2, if n ≡ 2 (mod 4),
2fn−1 + fn−2, if n ≡ 3 (mod 4),
3fn−1 + fn−2, if n ≡ 0 (mod 4).
Setting
gq+1 = (
√
12)q Uq
(
53
2
√
12
)
,
Z. Du et al.: New strong divisibility sequences 11
for q ⩾ 0, the first terms are:
n gn
1 1
2 53
3 2797
4 147605
5 7789501
6 411072293
7 21693357517
8 1144815080885
9 60414878996701
10 3188250805854533
Straightforward verification shows, for example, that g4 | g8 or g5 | g10. Furthermore,
gcd(g8, g10) = g2 or gcd(g6, g9) = g3,
and, of course,
gcd(g4, g9) = g1.
Finally, we study
fn =

2fn−1 + 4fn−2, if n ≡ 1 (mod 4),
fn−1 + 2fn−2, if n ≡ 2 (mod 4),
2fn−1 + fn−2, if n ≡ 3 (mod 4),
3fn−1 + fn−2, if n ≡ 0 (mod 4).
Setting
gq+1 = (
√
8)q Uq
(
46
2
√
8
)
,
for q ⩾ 0, the first terms are:
n gn
1 1
2 46
3 2108
4 96600
5 4426736
6 202857056
7 9296010688
8 425993635200
9 19521339133696
10 894573651068416
Now we can check, for instance, that g4 | g8 or g5 | g10. However,
gcd(g8, g10) = 2944.
12 Ars Math. Contemp. 22 (2022) #P1.08
ORCID iDs
Zhibin Du https://orcid.org/0000-0001-5795-3580
Darko Dimitrov https://orcid.org/0000-0002-1648-9600
Carlos M. da Fonseca https://orcid.org/0000-0001-7742-4416
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