Ellipse Chains and Associated Sequences

Hacène Belbachir and Soumeya Merwa Tebtoub Department of Mathematics USTHB, RECITS Laboratory

P. O. Box 32 El Alia 16111 Bab Ezzouar Algiers Algeria

hbelbachir@usthb.dz, hacenebelbachir@gmail.com
stebtoub@usthb.dz, tebtoubsoumeya@gmail.com

László Németh Institute of Mathematics

University of Sopron Bajcsy Zs. u. 4 Sopron H9400 Hungary and

USTHB, RECITS Laboratory

Algeria

nemeth.laszlo@uni-sopron.hu

Abstract

We define circle and ellipse chains tangent to the branches of a hyperbola and the terms of the chains are mutually tangent to each other. Our goal is to derive recurrence relations for the parameters of chains elements and to establish some connections between integer sequences and chains.

1 Introduction

Let us consider the hyperbola H with the canonical equation

\frac{x^{2}}{a^{2}} − \frac{y^{2}}{b^{2}} = 1,  (1)

and foci (±c, 0), where a and b are positive real numbers and c^{2} = a^{2}+b^{2}. Lucca [3] examined a tangential chain of circles inside the branch x > 0 of the hyperbola so that the n-th circle with center (x_{n}, 0) and radius r_{n} are tangent to the hyperbola and mutually tangent to each other. He showed that for certain a and b, the sequences (x_{n}/x_{0})_{n=0}^{∞} and (r_{n}/r_{0})_{n=0}^{∞} are integers. Belbachir et al. [2] extended the circle chains to ellipse chains inside the branch of hyperbola when the ratio of the minor and major axes is fixed. They described the recurrence relations for the ellipses’ parameters and determined a connection between the parameters of the ellipse chains and integer sequences.

In the present paper, we give an extension of the papers [2, 3]. We examine special chains of circles and ellipses between the branches of hyperbola H (or outside H), such that the circles (ellipses) are tangent to the hyperbola H and mutually tangent to each other. Furthermore, we give the recurrence relations for the tangential points. We define a tangential chain of ellipses between the branches of H where the centers of the ellipses coincide with the centers of the circles. We give recurrence relations for the ellipses’ parameters.

Our other main purpose is to give integer sequences that describe the parameters of our chains. We find more than fifty such integer sequences that appear in the On-Line Encyclopedia of Integer Sequences (OEIS) [4], and in this way, our investigation gives them geometrical interpretations. In what follows, we define t := a/b, s := b/a, λ := 2t^{2} + 1, and μ := t^{2} + 1.

2 Circle chains between the branches of hyperbola

Let us define a chain of circles with the following properties.

The canonical equation of the n-th circle centered at point (0, y_{n}), (n ≥ 0, y_{n} ≥ 0) is

x^{2} + (y − y_{n})^{2} = r_{n}^{2},  (2)

where the center of each circle lies on the y-axis, and r_{n} > 0 is the radius (Figure 1). The circles (2) are tangent to the hyperbola (1) and are mutually tangent, i.e.,

y_{n} − y_{n−1} = r_{n} + r_{n−1} (n ≥ 1).

Let (x̂_{n}, ŷ_{n}) be the tangential point of the branch x > 0 of hyperbola H and the n-th circle of the chain, given by ŷ_{n} = s^{2}/(1 + s^{2}) y_{n} and x̂_{n}^{2} = 1/(t^{2} + s^{2} + 2) y_{n}^{2} + a^{2}. Moreover

x̂_{n}^{2} = t^{2} ŷ_{n}^{2} + a^{2}.  (3)

The sequences (y_{n})_{n≥0}, (r_{n})_{n≥0}, (x̂_{n})_{n≥0} , and (ŷ_{n})_{n≥0} satisfy the following recurrence relations:

Theorem 1. The sequences (y_{n})_{n≥0}, (r_{n})_{n≥0}, and (ŷ_{n})_{n≥0} are second-order linear homogeneous recurrence sequences

ℓ_{n} = 2λℓ_{n−1} − ℓ_{n−2}  (n ≥ 2),  (4)

and the sequence (x̂_{n}^{2})_{n≥0} is a third-order linear homogeneous recurrence sequence

x̂_{n}^{2} = (4λ^{2} − 1) x̂_{n−1}^{2} − (4λ^{2} − 1) x̂_{n−2}^{2} + x̂_{n−3}^{2}  (n ≥ 3),  (5)

and the initial values are y_{0} = 0, r_{0} = a, x̂_{0}^{2} = a^{2}, ŷ_{0} = 0, y_{1} = 2aμ, r_{1} = aλ, x̂_{1}^{2} = a^{2}(4t^{2}+1), ŷ_{1} = 2a, and x̂_{2}^{2} = a^{2}(16t^{2}λ^{2} + 1).

Proof. The system composed of the equations (1) and (2) gives

\{ \begin{matrix}y_{n} = λ y_{n−1} + 2μ r_{n−1}, \\ r_{n} = 2t^{2} y_{n−1} + λ r_{n−1}. \end{matrix}

Then (y_{n})_{n≥0} and (r_{n})_{n≥0} satisfy (4) and this is also the case for (ŷ_{n})_{n≥0}, as y_{n} = μ ŷ_{n}. Now for x̂_{n}^{2}, let C = 1/(t^{2} + s^{2} + 2). Then x̂_{n}^{2} = Cy_{n}^{2} + a^{2}, y_{n} = 2λy_{n−1} − y_{n−2}, substituting y_{n} in x̂_{n}^{2} we get x̂_{n}^{2} = C(2λy_{n−1} − y_{n−2})^{2} + a^{2} = 4λ^{2}x̂_{n−1}^{2} + x̂_{n−2}^{2} − 4Cλy_{n−1}y_{n−2} − 4λ^{2}a^{2} and from the sum x̂_{n}^{2} + x̂_{n−1}^{2}, we obtain the equation (5).

Remark 2. Because of the equation (3), the recurrence relation for the sequence (ŷ_{n}^{2})_{n≥0} is the same as for the recurrence of (x̂_{n}^{2})_{n≥0}. Thus, the sequence of squared distances of tangential points and the origin (d_{n}^{2} = x̂_{n}^{2} + ŷ_{n}^{2})_{n≥0} satisfies the recurrence (5) as well. In the last section, we give a second-order recurrence solving (5).

3 Ellipse chains between the branches of hyperbola

In this section, we define a tangential chain of ellipses between the branches of H, where the ellipses’ centers coincide with the circles’ centers. Let us define a chain of ellipses with the following properties.

The canonical equation of the n-th ellipse centered at point (0, y_{n}) (y_{n} > 0, n ≥ 0) is

\frac{x^{2}}{α_{n}^{2}} + \frac{(y − y_{n})^{2}}{β_{n}^{2}} = 1,  (6)

where 2α_{n} > 0 is the width and 2β_{n} > 0 is the height of the ellipse (Figure 1).

Figure 1: Ellipse and circle chains between the branches of hyperbola.

Let the equation of the 0-th ellipse be x^{2}/a^{2} + y^{2}/β_{0}^{2} = 1, and thus y_{0} = 0, α_{0} = a. This implies that 0β_{0} < r_{0} + r_{1} = 2aμ.

The ellipses (6) are tangent to the hyperbola (1) and are mutually tangent, i.e.,

y_{n} − y_{n−1} = β_{n} + β_{n−1} (n ≥ 1).  (7)

The sequence y_{n} is the same as the sequence of circle chains. However, we can give recurrence formulas for the parameter β_{n} of ellipses (10). Let us determine α_{n}^{2}. From the system composed by the equations (1) and (6) and from the tangency condition between the ellipses, we have

(\frac{a^{2}}{b^{2}} + \frac{α_{n}^{2}}{β_{n}^{2}} ) y^{2} − \frac{2α_{n}^{2}}{β_{n}^{2}} y_{n}y + \frac{α_{n}^{2}}{β_{n}^{2}} y_{n}^{2} + a^{2} − α_{n}^{2} = 0.  (8)

Since the discriminant of equation (8) is equal to zero and after simplification, we get

s^{2}α_{n}^{4} − (b^{2} + y_{n}^{2} − β_{n}^{2}) α_{n}^{2} − a^{2}β_{n}^{2} = 0.  (9)

We put δ_{n} = α_{n}^{2} and ω_{n} = b^{2} + y_{n}^{2} − β_{n}^{2}. Then the solutions of the equation (9) are δ_{n,1} = (ω_{n} + \sqrt{ω_{n}^{2} + 4b^{2}β_{n}^{2}})/2s^{2} and δ_{n,2} = (ω_{n} − \sqrt{ω_{n}^{2} + 4b^{2}β_{n}^{2}})/2s^{2}. Since δ_{n,2} < 0, we get α_{n}^{2} = t^{2}/2(ω_{n} + \sqrt{ω_{n}^{2} + 4b^{2}β_{n}^{2}}). Let (x̃_{n}, ỹ_{n}) be the tangential point of the branch x > 0 of H and the n-th ellipse of the chain. Then recurrence relations for x̃_{n} and ỹ_{n} are as follows: ỹ_{n} = (α_{n}^{2}b^{2}/a^{2}β_{n}^{2} + α_{n}^{2}b^{2})y_{n} and x̃_{n}^{2} = (α_{n}^{4}a^{2}b^{2}/(a^{2}β_{n}^{2} + α_{n}^{2}b^{2})^{2})y_{n}^{2} + a^{2}, where y_{n}^{2} = (α_{n}^{2} − a^{2})(a^{2}β_{n}^{2} + α_{n}^{2}b^{2})/a^{2}α_{n}^{2}.

The following theorem gives the recurrence relations for (β_{n})_{n≥0}, (ỹ_{n})_{n≥0}, and (x̃_{n}^{2})_{n≥0}.

Theorem 3. The sequence (β_{n})_{n≥0} is a third-order linear homogeneous recurrence sequence

β_{n} = (2λ − 1) β_{n−1} + (2λ − 1) β_{n−2} − β_{n−3} (n ≥ 3).  (10)

The sequence (ỹ_{n})_{n≥0} satisfies the second-order linear homogeneous recurrence sequence (4) and the sequence (x̃_{n}^{2})_{n≥0} is a third-order linear homogeneous recurrence sequence

x̃_{n}^{2} = (4λ^{2} − 1) x̃_{n−1}^{2} − (4λ^{2} − 1) x̃_{n−2}^{2} + x̃_{n−3}^{2}  (n ≥ 3).  (11)

The initial values are β_{0}, β_{1} = 2aμ − β_{0}, β_{2} = 8at^{2}μ + β_{0}, ỹ_{0} = 0, x̃_{0}^{2} = a^{2},

ỹ_{1} = \frac{α_{1}(2 + 2a^{2})}{aβ_{1}^{2} + α_{1}^{2}b^{2}},
x̃_{1}^{2} = \frac{α_{1}^{8}a^{2}b^{2}(2a^{2} + 2)^{2}}{(a^{2}β_{1}^{2} + α_{1}^{2}b^{2})^{2}(aβ_{1}^{2} + α_{1}^{2}b^{2})^{2}} + a^{2}, and
x̃_{2}^{2} = \frac{α_{2}^{2}b^{2}(a^{2} + α_{2}^{2})(−a^{2}b^{2} + α_{2}^{2} + (b^{2} − β_{2}^{2})x_{2}^{2})}{(a^{2}β_{2}^{2} + α_{2}^{2}b^{2})^{2}} + a.

Proof. We have y_{n} is defined as follows: y_{n} = λ y_{n−1} + 2μ r_{n−1} where r_{n} is the radius of the circles. Using the tangency condition (7) we get

β_{n} = 2t^{2} y_{n−1} + 2μ r_{n−1} − β_{n−1}.  (12)

After calculations, we obtain β_{n} = 4t^{2} y_{n−1} + β_{n−2}. Subtracting β_{n−1} from β_{n} we obtain β_{n} − β_{n−1} = 4t^{2}(y_{n−1} − y_{n−2}) + β_{n−2} − β_{n−3}. Hence from Eq. (7), we find (10). The initial values come from (12). The proofs of the second-order and third-order linear homogeneous recurrence sequences of (ỹ_{n})_{n≥0} and (x̃_{n}^{2})_{n≥0} are similar to the proof of Theorem 1. Now C = α_{n}^{4}a^{2}b^{2}/(a^{2}β_{n}^{2} + α_{n}^{2}b^{2})^{2}.

Equations (5) and (11) are of the form V_{n} = (θ − 1)V_{n−1} − (θ − 1)V_{n−2} + V_{n−3} (n ≥ 3), and thus W_{n} = V_{n} − V_{n−1} is a second-order linear recurrence W_{n} = (θ − 2)W_{n−1} − W_{n−2} with θ = 4λ^{2}. We deduce an explicit form for (V_{n})_{n≥0} using (W_{n})_{n≥0} in the following theorem.

Theorem 4. For all n ≥ 2, V_{n} = V_{0} + ∑_{k=1}^{n} W_{k} .

Proof. The proof is left to the reader; it can be shown with some calculations. The explicit terms of the W_{k} are well known.

4 Associated integer sequences of chains

In this section, we determine conditions to relate the circle and ellipse chains with integer sequences. In what follow, let β_{0} = b.

4.1 Rectangular hyperbolas

Corollary 5. If H is a rectangular hyperbola (a = b), then r_{n} = α_{n} = β_{n}, ŷ_{n} = ỹ_{n}, and x̂_{n} = x̃_{n} for all non-negative integers n. Moreover, the recurrence sequence of (r_{n})_{n≥0}, (y_{n})_{n≥0}, (ŷ_{n})_{n≥0}, and (x̂_{n}^{2})_{n≥0} are, respectively, ℓ_{n} = 6ℓ_{n−1} −ℓ_{n−2} (n ≥ 2), with initial values r_{0} = a, r_{1} = 3a, y_{0} = 0, y_{1} = 4a, ŷ_{0} = 0, and ŷ_{1} = 2a and x̂_{n}^{2} = 35x̂_{n−1}^{2} − 35x̂_{n−2} + x̂_{n−3}^{2} (n ≥ 3), with initial values x̂_{0}^{2} = a^{2}, x̂_{1}^{2} = 5a^{2}, and x̂_{2}^{2} = 145a^{2}.

For rectangular hyperbolas H, from Corollary 5, we deduce the following result:

Theorem 6. If H is a rectangular hyperbola, then (r_{n})_{n≥0}, (y_{n})_{n≥0}, (ŷ_{n})_{n≥0}, and (x̂_{n}^{2})_{n≥0} are integer sequences, respectively, if and only if a is a positive integer, a = k/4, a = k/2, and a^{2} = k, respectively, where k is any positive integer.

Now we give some examples of integer sequences. Some of them appear in the OEIS [4] for a = 1: (r_{n})_{n≥0} = \underline{A001541}, (y_{n})_{n≥0} = \underline{A005319}, (ŷ_{n})_{n≥0} = \underline{A001542}, and (x̂_{n}^{2})_{n≥0} = \{0, \underline{A076218}\} and for general a with the conditions of Theorem 6, (r_{n})_{n≥0} = a · \underline{A001541}, (y_{n})_{n≥0} = a·\underline{A005319} = 2a·\underline{A001542}, (ŷ_{n})_{n≥0} = a·\underline{A001542}, and (x̂_{n}^{2})_{n≥0} = a^{2}·\{0, \underline{A076218}\}. For more sequences, see Table 1.

We mention that among the sequences in Table 1 and Table 5, there are some sequences, e.g., \underline{A098706}, which are defined without any combinatorial or geometrical interpretation. We provide a geometric interpretation. We also note that k · (y_{n}) = 2k · (ŷ_{n}).

Remark 7. The sequence (\underline{A001109}), e.g., appears, as (ŷ_{n})_{n≥0} and (y_{n})_{n≥0} when a = 1/2 and a = 1/4, dealing with balancing numbers [1].

a = b  (r_{n})_{n≥0}  (y_{n})_{n≥0}  (ŷ_{n})_{n≥0}  (x̂_{n}^{2})_{n≥0}
1  \underline{A001541}  \underline{A005319}  \underline{A001542}  \underline{A076218}
a  a·\underline{A001541}  a·\underline{A005319}  a·\underline{A001542}  a^{2}·\underline{A076218}
2  \underline{A003499}  \underline{A081554}  \underline{A005319}  \{4, 20, 580, 19604, …\}
3  \underline{A106329}  \{0, 12, 72, 420, …\} \underline{A075848} \{9, 45, 1305, 44109, …\}
4  \{4, 12, 68, 396, …\} \{0, 16, 96, 560, …\} \underline{A081554} \{16, 80, 2320, 78416, …\}
\sqrt{2}  –  –  –  \underline{A098706}\\{0\}

Table 1: Integer sequences connected to the regular hyperbola.

4.2 Integer sequences associated with circle chains

In order to generate only integer sequences, we state the following theorem.

Theorem 8. If t = (\sqrt{k − 2})/2, k ≥ 3, sequence (r_{n})_{n≥0}, a ∈ ℕ^{+} and a · k is even; (y_{n})_{n≥0}, a(k + 2) is even; (ŷ_{n})_{n≥0}, 2a is a positive integer, then b = 2a/\sqrt{k − 2} and the sequences (r_{n})_{n≥0}, (y_{n})_{n≥0}, and (ŷ_{n})_{n≥0} are integer sequences.

Proof. For (r_{n})_{n≥0}, we have r_{0} = a, so a ∈ ℕ^{+} and r_{1} = a(2t^{2} + 1) is integer if and only if (2t^{2} + 1) = m/a, where m is a suitable positive integer. From this, we find t^{2} = (m/a − 1)/2. For the first coefficient of (4), we have (4t^{2} + 2) = 2m/a = k where k is a suitable positive integer. Now m = (ak)/2 implies that ak is even and t^{2} = (k − 2) /4. Moreover, we have t = \sqrt{k − 2}/2, where t is positive if and only if k ≥ 3. Obviously, b = a/t comes from the definition of t.

In the case where (y_{n})_{n≥0}, let m = y_{1} = 2a(t^{2} + 1) be an integer. Then t^{2} = m/(2a) − 1 and t = \sqrt{m/(2a) − 1}, where m > 2a and a is a positive real number. From the coefficient (4t^{2} + 2) = k with a suitable positive integer, we have m = a(k + 2)/2. This implies that a(k + 2) is even and t^{2} = (k − 2)/4, k ≥ 3. Moreover, t = \sqrt{k − 2}/2 and b = a/t.

For (ŷ_{n})_{n≥0}, ŷ_{1} = 2a = m and (4t^{2}+2) = k, we obtain that 2a is even and t = \sqrt{k − 2}/2, k ≥ 3.

Theorem 9. If a^{2} is an integer and t = k/2, k ≥ 1, then the sequence (x̂_{n}^{2})_{n≥0} consists of integers and b = a/t.

Proof. All the coefficients and initial values of (5) are integers.

Table 1 and Table 5 (t = 1), moreover, Table 2, Table 3, and Table 4 contain examples of integer sequences. For a = 1 and b = 2, then (y_{n} − r_{n})_{n≥1} is the bisection of Lucas sequence \underline{A002878}.

\begin{matrix}a & b & t & (x̂_{n}^{2})_{n≥0} \\ \sqrt{2} & 1 & \sqrt{2} & \{2, 18, 1602, …\} \\ \sqrt{2} & 2 & ^{\sqrt{2}}/2 & \{2, 6, 66, 902, …\} \\ \sqrt{2} & ^{\sqrt{2}}/2 & 2 & \{2, 34, 10370, …\} \\ \sqrt{2} & 2\sqrt{2} & ^{1}/2 & \{2, 4, 20, 130, …\} \end{matrix}

Table 2: Integer sequences associated with circle chains.

a b t  (r_{n})_{n≥0}  (y_{n})_{n≥0}  (ŷ_{n})_{n≥0}  (x̂_{n}^{2})_{n≥0}
1 2 ^{1}/2  —  —  \{0, \underline{A025169}\};\underline{A111282},n ≥ 1  \underline{A064170}
1 ^{1}/2 2  \underline{A023039}  \{0, 10, 180, …\}  \underline{A207832}  \{1, 17, 5185, …\}
1 ^{1}/3 3  \underline{A078986}  \{0, 20, 760, …\}  \{0, 2, 76, 2886, …\}  \{1, 37, 51985, …\}
1 ^{2}/3 ^{3}/2  –  –  {0, 2, 22, 240 …}  {1, 10, 1090, …}
1 ^{1}/4 4  \underline{A099370}  \{0, 34, 2244, …\}  \{0, 2, 132, 8710, …\}  \{1, 65, 278785, …\}
2 1 2  \underline{A087215}  \underline{A004292}, n ≥ 1  \underline{A060645}  \{4, 68, 20740, …\}
2 4 ^{1}/2  \underline{A005248}  \underline{A201157}  \{0, 4, 12, 32, …\}  \{4, 8, 40, 260, …\}
2 ^{1}/2 4 \{2, 66, 4354, …\} \underline{A004298}, n ≥ 1  \{0, 4, 264, 17420, …\}  \{4, 260, 1115140, …\}
2 ^{1}/3 6 {2, 146, 21314, …} {0, 148, 21608, …} {0, 4, 584, 85260, …}  {4, 580, 12278020, …}
2 ^{2}/3 3 \{2, \underline{A239364}\} \{0, 13, 143, …\}  \{0, 4, 44, 480, …\}  \{4, 148, 207940, …\}
2 ^{4}/3 ^{3}/2  \underline{A057076}  \{0, 40, 1520, …\}  \{0, 4, 152, 5772, …\}  \{4, 40, 4360, …\}
3 1 3 \{3, 57, 2163, …\} \{0, 60, 2280, …\}  \underline{A084070}  \{9, 333, 467865, …\}
3 6 ^{1}/2  —  —  \underline{A099857}, n ≥ 1  \{9, 18, 90, 585, …\}
4 1 4 {4, 132, 8708, …} {0, 136, 8976, …}  {0, 8, 528, 34840, …} {16, 1040, 4460560, …}
4 2 2 \{4, 36, 644, …\} \{0, 40, 720, …\}  \underline{A134492}  \{16, 272, 82960, …\}
4 8 ^{1}/2 {4, 6, 14, 36, …} {0, 10, 30, 80, …}  {0, 8, 24, 64, …}  {16, 32, 160, …}
1 \sqrt{2} ^{\sqrt{2}}/2  \underline{A001075}  \underline{A005320}  \underline{A052530}  \underline{A011922}
1 ^{\sqrt{2}}/2 \sqrt{2}  \underline{A001079}  \underline{A122653}  \underline{A001078}  \{1, 9, 801, 78409, …\}
2 \sqrt{2} \sqrt{2}  \underline{A087799}  \underline{A004291}, n ≥ 1  \underline{A122652}  \{4, 36, 3204, …\}
2 ^{\sqrt{2}}/2 2\sqrt{2} \{2, 34, 1154, …\} \underline{A004294}, n ≥ 1  \underline{A202299}  \{4, 132, 147972, …\}
2 2\sqrt{2} ^{\sqrt{2}}/2  \underline{A003500}  \underline{A001352}, n ≥ 1  \underline{A231896}  \{4, 12, 132, …\}

Table 3: Integer sequences associated with circle chains.

a b  t  (y_{n})_{n≥0}  (ŷ_{n})_{n≥0}
1/2 1 ^{1}/2  —  \underline{A001906}
1/2 ^{1}/3  ^{3}/2  —  \{0, \underline{A004190}\}
1/2 ^{1}/4 2  \{0, 5, 90, …\}  \underline{A049660}
1/2 ^{1}/5  ^{5}/2  —  \{0,\underline{A049660}\}
1/2 ^{1}/6 3 \{0, 10, 380, …\} \{0, \underline{A078987}\}
1/2 ^{1}/7 ^{7}/2 — \underline{A097836}
1/2 ^{1}/8 4 \{0, 17, 1122, …\} \{0, \underline{A097316}\}
1/2 ^{1}/9 ^{9}/2 — \{0, \underline{A097839}\}
1/2 ^{1}/10 5 \{0, 26, 2652, …\} \{0, \underline{A097725}\}
1/2 ^{1}/11 ^{11}/2  —  \{0, \underline{A049670}\}
1/2 ^{1}/13 ^{13}/2  —  \{0, \underline{A097844}\}

Table 4: Integer sequences associated with circle chains.

a = b  (y_{n})_{n≥0}  (ŷ_{n})_{n≥0}
1/2  \underline{A001542}  \underline{A001109}
3/2  \underline{A075848}  \underline{A106328}
5/2 \{0, 10, 60, 350, …\} \underline{A276598}
7/2  \{0, \underline{A273182}\} \underline{A054890},n ≥ 1
9/2 \{0, 18, 108, 630, …\} \underline{A276602}
1/4  \underline{A001109}  –
3/4  \underline{A106328}  –
5/4  \underline{A276598}  –
7/4 \underline{A054890}, n ≥ 1  –

Table 5: Integer sequences connected to the regular hyperbola.

4.3 Integer sequences associated with ellipse chains

Theorem 10. For positive integers a, b, if b divides a then the sequence (β_{n})_{n≥0} is an integer sequence and its recurrence is

β_{n} = (4t^{2} + 1)β_{n−1} + (4t^{2} + 1)β_{n−2} − β_{n−3} (n ≥ 3),  (13)

and the initial values are β_{0} = b, β_{1} = 2aμ − b, and β_{2} = 8at^{2}μ + b.

Proof. We notice that the initial values of the sequence are integers. Thus, the coefficients of recurrence relation (13) are also integers, which guarantees that all the other terms of the sequence are integers.

Equations (10) and (13) are of the form V_{n} = (θ −1)V_{n−1} +(θ −1)V_{n−2} −V_{n−3}, (n ≥ 3), and thus W_{n} = V_{n} + V_{n−1} is a second-order linear recurrence W_{n} = θW_{n−1} − W_{n−2}. Here θ = 2λ and 4t^{2}, respectively. We deduce an explicit form for (V_{n})_{n≥0} using (W_{n})_{n≥0} and leaving the proof to the reader in the following theorem:

Theorem 11. For all n ≥ 2, we have V_{n} = (−1)^{n}V_{0} + ∑_{k=1}^{n}(−1)^{n−k}W_{k}.

The explicit terms of W_{k} are well known.

We give some integer recurrence sequences for (β_{n})_{n≥0} in Table 6. If t = 1, so a = b, then our hyperbola is a rectangular hyperbola, and it holds for the integer sequences associated with ellipse chain not only for the sequence (β_{n})_{n≥0}, but also for the sequences α_{n}, β_{n}, ỹ_{n}, and x̃_{n}. See Subsection 4.1.

\begin{matrix}a & b & t & (β_{n})_{n≥0} \\ a & a & 1 & in Table 1 \\ 2 & 1 & 2 & \{1, 19, 321, 5779, 103681, 1860499, …\} \\ 3 & 1 & 3 & \{1, 59, 2161, 82139, 3119041, 118441499, …\} \\ 4 & 1 & 4 & \{1, 135, 8705, 574599, 37914625, 2501790855, …\} \\ 4 & 2 & 2 & \{2, 38, 642, 11558, 207362, 3720998, …\} \end{matrix}

Table 6: Integer sequences associated with ellipse chains.

5 Acknowledgments

For H. B. and S. M. T., this work was supported by the grand of DGRSDT, number C0656701. For L. N. this work has been performed under the auspices of the program “Efop-3.6.1-16-2016-00018—Improving the role of the research + development + innovation in the higher education through institutional developments assisting intelligent specialization in Sopron and Szombathely”.

References

[1] M. Alp, N. Irmak, and L. Szalay, Balancing diophantine triples with distance 1, Period. Math. Hung. 71 (2015), 1–10.

[2] H. Belbachir, L. Németh, and S. M. Tebtoub, Integer sequences and ellipse chains inside a hyperbola, Ann. Math. Inform. 52 (2020), https://doi.org/10.33039/ami.2020.

   06.002.

[3] G. Lucca, Integer sequences and circle chains inside a hyperbola, Forum Geom. 19 (2019), 11–16.

[4] N. J. A. Sloane et al., The On-Line Encyclopedia of Integer Sequences, https://oeis.

   org.

2010 Mathematics Subject Classification: Primary 11B37; Secondary 52C26. Keywords: circle chain, ellipse chain, ellipsoid chain, recurrence relation, integer sequence.

(Concerned with sequences \underline{A001075}, \underline{A001078}, \underline{A001079}, \underline{A001109}, \underline{A001352}, \underline{A001541}, \underline{A001542}, \underline{A001906}, \underline{A002878}, \underline{A003499}, \underline{A003500}, \underline{A004190}, \underline{A004291}, \underline{A004292}, \underline{A004294}, \underline{A004298}, \underline{A005248}, \underline{A005319}, \underline{A005320}, \underline{A011922}, \underline{A023039}, \underline{A025169}, \underline{A049660}, \underline{A049670}, \underline{A052530}, \underline{A054890}, \underline{A057076}, \underline{A060645}, \underline{A064170}, \underline{A075848}, \underline{A076218}, \underline{A078986}, \underline{A078987}, \underline{A081554}, \underline{A084070}, \underline{A087215}, \underline{A087799}, \underline{A097316}, \underline{A097725}, \underline{A097836}, \underline{A097839}, \underline{A097844}, \underline{A098706}, \underline{A099370}, \underline{A099857}, \underline{A106328}, \underline{A106329}, \underline{A111282}, \underline{A122652}, \underline{A122653}, \underline{A134492}, \underline{A201157}, \underline{A202299}, \underline{A207832}, \underline{A231896}, \underline{A239364}, \underline{A273182}, \underline{A276598}, and \underline{A276602}.)

Received April 8 2020; revised versions received August 27 2020; September 10 2020. Published in Journal of Integer Sequences, September 11 2020.

Return to Journal of Integer Sequences home page.