# The sequences of hexagonal lattice based equal circle packing inside rectangles on the Euclidian plane

Jure Voglar

<figure><img src="image_1.png" /></figure>

## To cite this version:

Jure Voglar. The sequences of hexagonal lattice based equal circle packing inside rectangles on the Euclidian plane. 2024. hal-04552052

HAL Id: hal-04552052

https://hal.science/hal-04552052

Preprint submitted on 19 Apr 2024

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

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<figure><img src="image_2.png" /></figure>

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# The sequences of hexagonal lattice based equal circle packing inside rectangles on the Euclidian plane

Jure Voglar

<figure><img src="image_1.png" /></figure>

## To cite this version:

Jure Voglar. The sequences of hexagonal lattice based equal circle packing inside rectangles on the Euclidian plane. 2024. hal-04552052

HAL Id: hal-04552052

https://hal.science/hal-04552052v1

Preprint submitted on 19 Apr 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# The sequences of hexagonal lattice based equal circle packing inside rectangles on the Euclidian plane

Jure Voglar<sup>a *</sup>

## Abstract

The mathematical sequences related to hexagonal lattice based equal circle packing inside rectangles with aspect ratios close to one on the Euclidian plane are derived, presented and discussed in the article. The circle packing problem is divided into two cases related to even and odd index numbers of the initial problem description, which correspond to even and odd numbers of rows of the packed circles. In both of the cases, there are multiple similarities. The sequences equal to relative lengths of the sides of the rectangles delimiting the packed circles are first-degree polynomials. Rational functions describe the aspect ratios and the packing densities. The numbers of packed circles are equal to second-degree polynomials. The packing density's limit values are in both cases equal to the value of the highest-density lattice packing of $\pi/(2\sqrt{3})$. Both packing densities increase with increasing index number.

## Keywords

circle packing; Euclidian plane; hexagonal lattice; mathematical sequences; rectangle

<footnote><sup>a</sup> Department of Catalysis and Chemical Reaction Engineering, National Institute of Chemistry, Hajdrihova 19, 1001 Ljubljana, Slovenia, email: jure.voglar@ki.si</footnote>

<footnote>* corresponding author</footnote>
# The sequences of hexagonal lattice based equal circle packing inside rectangles on the Euclidian plane

Jure Voglar<sup>a *</sup>

## Abstract

The mathematical sequences related to hexagonal lattice based equal circle packing inside rectangles with aspect ratios close to one on the Euclidian plane are derived, presented and discussed in the article. The circle packing problem is divided into two cases related to even and odd index numbers of the initial problem description, which correspond to even and odd numbers of rows of the packed circles. In both of the cases, there are multiple similarities. The sequences equal to relative lengths of the sides of the rectangles delimiting the packed circles are first-degree polynomials. Rational functions describe the aspect ratios and the packing densities. The numbers of packed circles are equal to second-degree polynomials. The packing density's limit values are in both cases equal to the value of the highest-density lattice packing of $\pi/(2\sqrt{3})$. Both packing densities increase with increasing index number.

## Keywords

circle packing; Euclidian plane; hexagonal lattice; mathematical sequences; rectangle

<footnote><sup>a</sup> Department of Catalysis and Chemical Reaction Engineering, National Institute of Chemistry, Hajdrihova 19, 1001 Ljubljana, Slovenia, email: jure.voglar@ki.si</footnote>

<footnote>* corresponding author</footnote>
# 1. Introduction

Investigations into the circle packing on the Euclidian plane have a long history. J.L. Lagrange proved (in 1773) that the highest-density lattice packing of equal circles on the Euclidean plane is the hexagonal packing arrangement, where the centers of circles are arranged in a hexagonal lattice and each circle is surrounded by six adjacent circles. It is simple to determine that this equal circle packing density equals to $\pi/(2\sqrt{3})$. A. Thue attempted to prove that this highest-density is optimal among all packings in 1890 [1]. L. F. Tóth published the first rigorous proof in 1942 [2].

Circle packing problems for equal circles in various geometries have been explored in multiple studies e.g. [3–7]. Different numerical algorithms were obtained and tested to achieve high circle packing densities in various different geometrically restricted surface areas [8–13]. While there are numerous works on numerical optimizations of the circle packing problems, there are hardly any recent advancements revealing and/or describing the new mathematical properties of the circle packing available in the literature.

Our previous work [14] revealed the mathematical sequences related to hexagonal lattice based equal circle packing inside bounding circles, equilateral triangles and equilateral hexagons. The current investigation is on the other hand focused on hexagonal lattice based equal circle packing inside rectangles with aspect ratios close to one on the Euclidian plane by derivation of the sequences related to the relevant geometrical features and the packing density. The presented sequences can be useful in multiple potential applications [15].

# 2. Methods

Firstly, we have manually drawn and derived the quantities (number of packed circles $N_i$, relative lengths of the sides of the rectangles $a_i$ and $b_i$) for the first 6 elements of the circle packing arrangements. The rectangles were selected in a manner that they bound a hexagonal lattice based packed equal circles with an equal value of the maximal number of circles packed in an individual row to a number of rows of the packed circles e.g. at $i = 4$ there are up to 4 circles packed into rows with the total number of rows equal to 4 (Figure 1). Therefore, the aspect ratios of the rectangles ($AR_i$) remain close (or equal to one at $i = 1$) to one, with the side $b_i$ being shorter (or equal to the side $a_i$ at $i = 1$) than the side $a_i$.
# 1. Introduction

Investigations into the circle packing on the Euclidian plane have a long history. J.L. Lagrange proved (in 1773) that the highest-density lattice packing of equal circles on the Euclidean plane is the hexagonal packing arrangement, where the centers of circles are arranged in a hexagonal lattice and each circle is surrounded by six adjacent circles. It is simple to determine that this equal circle packing density equals to $\pi/(2\sqrt{3})$. A. Thue attempted to prove that this highest-density is optimal among all packings in 1890 [1]. L. F. Tóth published the first rigorous proof in 1942 [2].

Circle packing problems for equal circles in various geometries have been explored in multiple studies e.g. [3–7]. Different numerical algorithms were obtained and tested to achieve high circle packing densities in various different geometrically restricted surface areas [8–13]. While there are numerous works on numerical optimizations of the circle packing problems, there are hardly any recent advancements revealing and/or describing the new mathematical properties of the circle packing available in the literature.

Our previous work [14] revealed the mathematical sequences related to hexagonal lattice based equal circle packing inside bounding circles, equilateral triangles and equilateral hexagons. The current investigation is on the other hand focused on hexagonal lattice based equal circle packing inside rectangles with aspect ratios close to one on the Euclidian plane by derivation of the sequences related to the relevant geometrical features and the packing density. The presented sequences can be useful in multiple potential applications [15].

# 2. Methods

Firstly, we have manually drawn and derived the quantities (number of packed circles $N_i$, relative lengths of the sides of the rectangles $a_i$ and $b_i$) for the first 6 elements of the circle packing arrangements. The rectangles were selected in a manner that they bound a hexagonal lattice based packed equal circles with an equal value of the maximal number of circles packed in an individual row to a number of rows of the packed circles e.g. at $i = 4$ there are up to 4 circles packed into rows with the total number of rows equal to 4 (Figure 1). Therefore, the aspect ratios of the rectangles ($AR_i$) remain close (or equal to one at $i = 1$) to one, with the side $b_i$ being shorter (or equal to the side $a_i$ at $i = 1$) than the side $a_i$.
<figure><img src="image_3.png" /><figcaption>Figure 1. The first 6 equal circle packing arrangements inside the rectangles, where *i* is the index number and *r* is the radius of the packed circles.</figcaption></figure>

After that, we extracted the mathematical sequences behind the circle packing inside rectangles and derived the equations for the packing densities.

### 3. Results and discussion

The Table 1 presents the values of the number of packed circles and relative (to the radius of the packed circles *r*) lengths of the sides of the rectangle.

<table><caption>Table 1. Values of the number of packed circles and relative lengths of the sides of the rectangle for the first 6 index numbers of the circle packing inside the rectangles presented in the Figure 1.</caption><thead><tr><th>i</th><th>N<sub>i</sub></th><th>a<sub>i</sub>/r</th><th>b<sub>i</sub>/r</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>2</td><td>2</td></tr><tr><td>2</td><td>3<br>(2+1)</td><td>4</td><td>2+√3</td></tr><tr><td>3</td><td>8<br>(3·2+2)</td><td>6</td><td>2+2·√3</td></tr><tr><td>4</td><td>14<br>(4·2+3·2)</td><td>8</td><td>2+3·√3</td></tr><tr><td>5</td><td>23<br>(5·3+4·2)</td><td>10</td><td>2+4·√3</td></tr><tr><td>6</td><td>33<br>(6·3+5·3)</td><td>12</td><td>2+5·√3</td></tr></tbody></table>
<figure><img src="image_3.png" /><figcaption>Figure 1. The first 6 equal circle packing arrangements inside the rectangles, where *i* is the index number and *r* is the radius of the packed circles.</figcaption></figure>

After that, we extracted the mathematical sequences behind the circle packing inside rectangles and derived the equations for the packing densities.

### 3. Results and discussion

The Table 1 presents the values of the number of packed circles and relative (to the radius of the packed circles *r*) lengths of the sides of the rectangle.

<table><caption>Table 1. Values of the number of packed circles and relative lengths of the sides of the rectangle for the first 6 index numbers of the circle packing inside the rectangles presented in the Figure 1.</caption><thead><tr><th>i</th><th>N<sub>i</sub></th><th>a<sub>i</sub>/r</th><th>b<sub>i</sub>/r</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>2</td><td>2</td></tr><tr><td>2</td><td>3<br>(2+1)</td><td>4</td><td>2+√3</td></tr><tr><td>3</td><td>8<br>(3·2+2)</td><td>6</td><td>2+2·√3</td></tr><tr><td>4</td><td>14<br>(4·2+3·2)</td><td>8</td><td>2+3·√3</td></tr><tr><td>5</td><td>23<br>(5·3+4·2)</td><td>10</td><td>2+4·√3</td></tr><tr><td>6</td><td>33<br>(6·3+5·3)</td><td>12</td><td>2+5·√3</td></tr></tbody></table>
The sequence of the number of packed circles (1, 3, 8, 14, 23, 33) could unfortunately not be identified and its equation extracted. On the other hand, the equations for the relative lengths of sides of the bounding rectangles were easily determined to be:

$$ \frac{a_i}{r} = 2i, $$

$$ \frac{b_i}{r} = 2 + (i - 1)\sqrt{3}. $$

The aspect ratio of the bounding rectangle is:

$$ AR_i = \frac{a_i}{b_i} = \frac{(2i)r}{(2+(i-1)\sqrt{3})r} = \frac{2i}{\sqrt{3i+(2-\sqrt{3})}} = \frac{2}{\sqrt{3}} + \frac{2-\frac{4}{\sqrt{3}}}{\sqrt{3i+(2-\sqrt{3})}}. $$

The aspect ratio's limit as the index number $i$ tends to infinity is:

$$ \lim_{i \to \infty} AR_i = \frac{2}{\sqrt{3}} \approx 1.1547. $$

The circle packing density can be calculated with equation:

$$ \rho_{p,i} = \frac{N_i \pi r^2}{a_i b_i}. $$

Two different cases (sequences) were identified for the number of packed circles in the Table 1. The first one included the even numbers of index $i$ (2, 4 and 6), while the second one included the odd ones ($i$ = 1, 3 and 5).

The sequence for even index numbers is:

$$ N_{i,even} = i\left(\frac{i}{2}\right) + (i-1)\left(\frac{i}{2}\right) = (2i-1)\left(\frac{i}{2}\right) = i^2 - \frac{i}{2}. $$

While the sequence for odd index numbers is:

$$ N_{i,odd} = i\left(\frac{i+1}{2}\right) + (i-1)\left(\frac{i-1}{2}\right) = \frac{i^2+i}{2} + \frac{i^2-2i+1}{2} = \frac{2i^2-i+1}{2} = i^2 - \frac{i}{2} + \frac{1}{2}. $$

The two cases were separately studied in the next two subsections of the article.

### 3.1. Sequences for even index numbers

The index numbers were replaced in a way that the new index numbers were half of the previous index numbers:

$$ i_{old} = 2i_{new}. $$

The index numbers in this subsection denoted with $i$ shall not be confused with the initial index numbers and index numbers in the section 3.2. They are presented in the Table 2.
The sequence of the number of packed circles (1, 3, 8, 14, 23, 33) could unfortunately not be identified and its equation extracted. On the other hand, the equations for the relative lengths of sides of the bounding rectangles were easily determined to be:

$$ \frac{a_i}{r} = 2i, $$

$$ \frac{b_i}{r} = 2 + (i - 1)\sqrt{3}. $$

The aspect ratio of the bounding rectangle is:

$$ AR_i = \frac{a_i}{b_i} = \frac{(2i)r}{(2+(i-1)\sqrt{3})r} = \frac{2i}{\sqrt{3i+(2-\sqrt{3})}} = \frac{2}{\sqrt{3}} + \frac{2-\frac{4}{\sqrt{3}}}{\sqrt{3i+(2-\sqrt{3})}} $$

The aspect ratio's limit as the index number $i$ tends to infinity is:

$$ \lim_{i \to \infty} AR_i = \frac{2}{\sqrt{3}} \approx 1.1547. $$

The circle packing density can be calculated with equation:

$$ \rho_{p,i} = \frac{N_i \pi r^2}{a_i b_i}. $$

Two different cases (sequences) were identified for the number of packed circles in the Table 1. The first one included the even numbers of index $i$ (2, 4 and 6), while the second one included the odd ones ($i$ = 1, 3 and 5).

The sequence for even index numbers is:

$$ N_{i,even} = i\left(\frac{i}{2}\right) + (i-1)\left(\frac{i}{2}\right) = (2i-1)\left(\frac{i}{2}\right) = i^2 - \frac{i}{2}. $$

While the sequence for odd index numbers is:

$$ N_{i,odd} = i\left(\frac{i+1}{2}\right) + (i-1)\left(\frac{i-1}{2}\right) = \frac{i^2+i}{2} + \frac{i^2-2i+1}{2} = \frac{2i^2-i+1}{2} = i^2 - \frac{i}{2} + \frac{1}{2}. $$

The two cases were separately studied in the next two subsections of the article.

### 3.1. Sequences for even index numbers

The index numbers were replaced in a way that the new index numbers were half of the previous index numbers:

$$ i_{old} = 2i_{new}. $$

The index numbers in this subsection denoted with $i$ shall not be confused with the initial index numbers and index numbers in the section 3.2. They are presented in the Table 2.
<table><caption>Table 2. Values of the number of packed circles and relative lengths of the sides of the rectangles for the first 3 index numbers of the circle packing inside the rectangles with even numbers of rows of the packed circles.</caption><thead><tr><th>i</th><th>N<sub>i</sub></th><th>a<sub>i</sub>/r</th><th>b<sub>i</sub>/r</th></tr></thead><tbody><tr><td>1</td><td>3<br>(2+1)</td><td>4</td><td>2+√3</td></tr><tr><td>2</td><td>14<br>(4·2+3·2)</td><td>8</td><td>2+3·√3</td></tr><tr><td>3</td><td>33<br>(6·3+5·3)</td><td>12</td><td>2+5·√3</td></tr></tbody></table>

The previous equation was simply inserted into the already known equations for determination of $a_i$, $b_i$, $AR_i$ and $N_i$, which yielded the new equations:

$$ \frac{a_i}{r} = 2(2i) = 4i, $$

$$ \frac{b_i}{r} = 2 + (2i - 1)\sqrt{3}, $$

$$ AR_i = \frac{2}{\sqrt{3}} + \frac{2-\frac{4}{\sqrt{3}}}{2\sqrt{3}i+(2-\sqrt{3})}, $$

$$ N_i = (2i)^2 - \frac{2i}{2} = 4i^2 - i. $$

We were then able to derive the sequence equal to the packing density:

$$ \rho_{p,i} = \frac{N_i \pi r^2}{a_i b_i} = \frac{(4i^2 - i) \pi r^2}{4i(2 + (2i - 1)\sqrt{3})r^2} = \frac{4\pi i^2 - \pi i}{8i + 8\sqrt{3}i^2 - 4\sqrt{3}i} = \frac{4\pi i^2 - \pi i}{8\sqrt{3}i^2 + (8 - 4\sqrt{3})i} = \frac{\pi}{2\sqrt{3}} + \frac{(1 - \frac{4}{\sqrt{3}})\pi i}{8\sqrt{3}i^2 + (8 - 4\sqrt{3})i} $$

The packing density's limit as the index number $i$ tends to infinity is:

$$ \lim_{i \to \infty} \rho_{p,i} = \frac{\pi}{2\sqrt{3}} \approx 0.9069. $$

This limit value is equal to the packing density of hexagonal lattice based equal circle packing on the Euclidian plane, which is presented in the introduction.

The sequences describing relative lengths of the sides of the rectangles of packed circles take form of different first-degree polynomials (with $i$ as the variable). The aspect ratio and the packing density are expressed as different rational functions. While the number of packed circles forms a second-degree polynomial.

## 3.2. Sequences for odd index numbers

As in the subsection 3.1, the index numbers were replaced; In this case following the relation:

$$ i_{old} = 2i_{new} - 1. $$
<table><caption>Table 2. Values of the number of packed circles and relative lengths of the sides of the rectangles for the first 3 index numbers of the circle packing inside the rectangles with even numbers of rows of the packed circles.</caption><thead><tr><th>i</th><th>N<sub>i</sub></th><th>a<sub>i</sub>/r</th><th>b<sub>i</sub>/r</th></tr></thead><tbody><tr><td>1</td><td>3<br>(2+1)</td><td>4</td><td>2+√3</td></tr><tr><td>2</td><td>14<br>(4·2+3·2)</td><td>8</td><td>2+3·√3</td></tr><tr><td>3</td><td>33<br>(6·3+5·3)</td><td>12</td><td>2+5·√3</td></tr></tbody></table>

The previous equation was simply inserted into the already known equations for determination of $a_i$, $b_i$, $AR_i$ and $N_i$, which yielded the new equations:

$$ \frac{a_i}{r} = 2(2i) = 4i, $$

$$ \frac{b_i}{r} = 2 + (2i - 1)\sqrt{3}, $$

$$ AR_i = \frac{2}{\sqrt{3}} + \frac{2-\frac{4}{\sqrt{3}}}{2\sqrt{3}i+(2-\sqrt{3})}, $$

$$ N_i = (2i)^2 - \frac{2i}{2} = 4i^2 - i. $$

We were then able to derive the sequence equal to the packing density:

$$ \rho_{p,i} = \frac{N_i \pi r^2}{a_i b_i} = \frac{(4i^2 - i) \pi r^2}{4i(2 + (2i-1)\sqrt{3})r^2} = \frac{4\pi i^2 - \pi i}{8i + 8\sqrt{3}i^2 - 4\sqrt{3}i} = \frac{4\pi i^2 - \pi i}{8\sqrt{3}i^2 + (8 - 4\sqrt{3})i} = \frac{\pi}{2\sqrt{3}} + \frac{(1 - \frac{4}{\sqrt{3}})\pi i}{8\sqrt{3}i^2 + (8 - 4\sqrt{3})i} $$

The packing density's limit as the index number $i$ tends to infinity is:

$$ \lim_{i \to \infty} \rho_{p,i} = \frac{\pi}{2\sqrt{3}} \approx 0.9069. $$

This limit value is equal to the packing density of hexagonal lattice based equal circle packing on the Euclidian plane, which is presented in the introduction.

The sequences describing relative lengths of the sides of the rectangles of packed circles take form of different first-degree polynomials (with $i$ as the variable). The aspect ratio and the packing density are expressed as different rational functions. While the number of packed circles forms a second-degree polynomial.

## 3.2. Sequences for odd index numbers

As in the subsection 3.1, the index numbers were replaced; In this case following the relation:

$$ i_{old} = 2i_{new} - 1. $$
The index numbers in this subsection denoted with *i* shall not be confused with the index numbers in the other parts of the article. The Table 3 contains the new index numbers with corresponding values of the number of packed circles and relative lengths of the sides of the rectangles.

<table><caption>Table 3. Values of the number of packed circles and relative lengths of the sides of the rectangles for the first 3 index numbers of the circle packing inside the rectangles with odd numbers of rows of the packed circles.</caption><thead><tr><th><i>i</i></th><th><i>N</i><sub><i>i</i></sub></th><th><i>a</i><sub><i>i</i></sub>/<i>r</i></th><th><i>b</i><sub><i>i</i></sub>/<i>r</i></th></tr></thead><tbody><tr><td>1</td><td>1</td><td>2</td><td>2</td></tr><tr><td>2</td><td>8<br>(3·2+2)</td><td>6</td><td>2+2·√3</td></tr><tr><td>3</td><td>23<br>(5·3+4·2)</td><td>10</td><td>2+4·√3</td></tr></tbody></table>

The last equation was inserted in the previously known equations for of $a_i$, $b_i$, $AR_i$ and $N_i$ to generate the equations:

$$ \frac{a_i}{r} = 2(2i - 1) = 4i - 2, $$

$$ \frac{b_i}{r} = 2 + (2i - 2)\sqrt{3}, $$

$$ AR_i = \frac{2}{\sqrt{3}} + \frac{2 - \frac{4}{\sqrt{3}}}{\sqrt{3}(2i-1) + (2-\sqrt{3})} = \frac{2}{\sqrt{3}} + \frac{2 - \frac{4}{\sqrt{3}}}{2\sqrt{3}i + (2-2\sqrt{3})}, $$

$$ N_i = (2i - 1)^2 - \frac{2i - 1}{2} + \frac{1}{2} = 4i^2 - 4i + 1 - i + 1 = 4i^2 - 5i + 2. $$

The packing density is equal to:

$$ \rho_{p,i} = \frac{N_i \pi r^2}{a_i b_i} = \frac{(4i^2-5i+2)\pi r^2}{(4i-2)(2+(2i-2)\sqrt{3})r^2} = \frac{4\pi i^2 - 5\pi i + 2\pi}{8i+8\sqrt{3}i^2 - 8\sqrt{3}i - 4 - 4\sqrt{3}i + 4\sqrt{3}} = \frac{4\pi i^2 - 5\pi i + 2\pi}{8\sqrt{3}i^2 + (8-12\sqrt{3})i + (4\sqrt{3}-4)} = \frac{\pi}{2\sqrt{3}} + \frac{(1-\frac{4}{\sqrt{3}})\pi i + \frac{2\pi}{\sqrt{3}}}{8\sqrt{3}i^2 + (8-12\sqrt{3})i + (4\sqrt{3}-4)}. $$

The packing density's limit as the index number *i* tends to infinity is the same as in the subsection 3.1:

$$ \lim_{i \to \infty} \rho_{p,i} = \frac{\pi}{2\sqrt{3}} \approx 0.9069. $$

The value of the packing density's limit is equal to its value in the subsection 3.1 and to the packing density of hexagonal lattice based equal circle packing on the Euclidian plane.
The index numbers in this subsection denoted with *i* shall not be confused with the index numbers in the other parts of the article. The Table 3 contains the new index numbers with corresponding values of the number of packed circles and relative lengths of the sides of the rectangles.

<table><caption>Table 3. Values of the number of packed circles and relative lengths of the sides of the rectangles for the first 3 index numbers of the circle packing inside the rectangles with odd numbers of rows of the packed circles.</caption><thead><tr><th><i>i</i></th><th><i>N</i><sub><i>i</i></sub></th><th><i>a</i><sub><i>i</i></sub>/<i>r</i></th><th><i>b</i><sub><i>i</i></sub>/<i>r</i></th></tr></thead><tbody><tr><td>1</td><td>1</td><td>2</td><td>2</td></tr><tr><td>2</td><td>8<br>(3·2+2)</td><td>6</td><td>2+2·√3</td></tr><tr><td>3</td><td>23<br>(5·3+4·2)</td><td>10</td><td>2+4·√3</td></tr></tbody></table>

The last equation was inserted in the previously known equations for of $a_i$, $b_i$, $AR_i$ and $N_i$ to generate the equations:

$$ \frac{a_i}{r} = 2(2i - 1) = 4i - 2, $$

$$ \frac{b_i}{r} = 2 + (2i - 2)\sqrt{3}, $$

$$ AR_i = \frac{2}{\sqrt{3}} + \frac{2 - \frac{4}{\sqrt{3}}}{\sqrt{3}(2i-1) + (2-\sqrt{3})} = \frac{2}{\sqrt{3}} + \frac{2 - \frac{4}{\sqrt{3}}}{2\sqrt{3}i + (2-2\sqrt{3})}, $$

$$ N_i = (2i - 1)^2 - \frac{2i - 1}{2} + \frac{1}{2} = 4i^2 - 4i + 1 - i + 1 = 4i^2 - 5i + 2. $$

The packing density is equal to:

$$ \rho_{p,i} = \frac{N_i \pi r^2}{a_i b_i} = \frac{(4i^2 - 5i + 2) \pi r^2}{(4i-2)(2 + (2i-2)\sqrt{3})r^2} = \frac{4\pi i^2 - 5\pi i + 2\pi}{8i + 8\sqrt{3}i^2 - 8\sqrt{3}i - 4 - 4\sqrt{3}i + 4\sqrt{3}} = \frac{4\pi i^2 - 5\pi i + 2\pi}{8\sqrt{3}i^2 + (8 - 12\sqrt{3})i + (4\sqrt{3} - 4)} = \frac{\pi}{2\sqrt{3}} + \frac{(1 - \frac{4}{\sqrt{3}})\pi i + \frac{2\pi}{\sqrt{3}}}{8\sqrt{3}i^2 + (8 - 12\sqrt{3})i + (4\sqrt{3} - 4)}. $$

The packing density's limit as the index number *i* tends to infinity is the same as in the subsection 3.1:

$$ \lim_{i \to \infty} \rho_{p,i} = \frac{\pi}{2\sqrt{3}} \approx 0.9069. $$

The value of the packing density's limit is equal to its value in the subsection 3.1 and to the packing density of hexagonal lattice based equal circle packing on the Euclidian plane.
The sequences equal to relative lengths of the sides of the rectangles of the packed circles are different first-degree polynomials. Different rational functions describe the aspect ratio and the packing density. The number of packed circles equals to a second-degree polynomial.

**4. Conclusions**

In the article we derived and presented the mathematical sequences related to hexagonal lattice based equal circle packing inside rectangles with aspect ratios close to one on the Euclidian plane. The results led us to the following conclusions:

* The selected circle packing arrangement inside rectangle unfortunately does not have a single identifiable mathematical sequence to describe the number of packed circles. The other relevant quantities were on the other hand easily extracted as sequences.

* We were able to identify two different sequences related to the number of packed circles; One related to even and the other to odd index numbers i.

* The sequences equal to relative lengths of the sides of the rectangles of the packed circles are different first-degree polynomials. Different rational functions describe the aspect ratios and the packing densities. The numbers of packed circles are equal to different second-degree polynomials. These notions are valid for both cases related to even and odd numbers of rows of the packed circles.

* The packing density's limit values are in both cases equal to the value of the highest-density lattice packing of $\pi/(2\sqrt{3})$. The packing densities increases with increasing index number i.

* The findings of this paper may have multiple practical applications related with advancements, decision-making and optimizations in design with many industrial, engineering and logistics applications.

* This article could encourage mathematicians to apply sequences to multiple similar geometrical problems.

**Acknowledgements**

The author thanks the support from the Slovenian Research and Innovation Agency (ARIS) through core funding P2-0152.

**References**

[1] H.C. Chang, L.C. Wang, A simple proof of Thue's theorem on circle packing, ArXiv Preprint ArXiv:1009.4322 (2010).

[2] L. Fejes, Über die dichteste Kugellagerung, Mathematische Zeitschrift 48 (1942) 676–684.
The sequences equal to relative lengths of the sides of the rectangles of the packed circles are different first-degree polynomials. Different rational functions describe the aspect ratio and the packing density. The number of packed circles equals to a second-degree polynomial.

**4. Conclusions**

In the article we derived and presented the mathematical sequences related to hexagonal lattice based equal circle packing inside rectangles with aspect ratios close to one on the Euclidian plane. The results led us to the following conclusions:

* The selected circle packing arrangement inside rectangle unfortunately does not have a single identifiable mathematical sequence to describe the number of packed circles. The other relevant quantities were on the other hand easily extracted as sequences.

* We were able to identify two different sequences related to the number of packed circles; one related to even and the other to odd index numbers i.

* The sequences equal to relative lengths of the sides of the rectangles of the packed circles are different first-degree polynomials. Different rational functions describe the aspect ratios and the packing densities. The numbers of packed circles are equal to different second-degree polynomials. These notions are valid for both cases related to even and odd numbers of rows of the packed circles.

* The packing density's limit values are in both cases equal to the value of the highest-density lattice packing of $\pi/(2\sqrt{3})$. The packing densities increases with increasing index number i.

* The findings of this paper may have multiple practical applications related with advancements, decision-making and optimizations in design with many industrial, engineering and logistics applications.

* This article could encourage mathematicians to apply sequences to multiple similar geometrical problems.

**Acknowledgements**

The author thanks the support from the Slovenian Research and Innovation Agency (ARIS) through core funding P2-0152.

**References**

[1] H.C. Chang, L.C. Wang, A simple proof of Thue's theorem on circle packing, ArXiv Preprint ArXiv:1009.4322 (2010).

[2] L. Fejes, Über die dichteste Kugellagerung, Mathematische Zeitschrift 48 (1942) 676–684.
[3] E.G. Birgin, J.M. Gentil, New and improved results for packing identical unitary radius circles within triangles, rectangles and strips, Comput Oper Res 37 (2010) 1318–1327.

[4] E. Specht, High density packings of equal circles in rectangles with variable aspect ratio, Comput Oper Res 40 (2013) 58–69.

[5] K.J. Nurmela, P.R.J. Östergård, Packing up to 50 equal circles in a square, Discrete Comput Geom 18 (1997) 111–120.

[6] P.G. Szabó, E. Specht, Packing up to 200 equal circles in a square, in: Models and Algorithms for Global Optimization: Essays Dedicated to Antanas Žilinskas on the Occasion of His 60th Birthday, Springer, 2007: pp. 141–156.

[7] P. Amore, D. de la Cruz, V. Hernandez, I. Rincon, U. Zarate, Circle packing in arbitrary domains, Physics of Fluids 35 (2023).

[8] C.O. López, J.E. Beasley, A heuristic for the circle packing problem with a variety of containers, Eur J Oper Res 214 (2011) 512–525.

[9] G.L. Orick, K. Stephenson, C. Collins, A linearized circle packing algorithm, Computational Geometry 64 (2017) 13–29.

[10] I. Castillo, F.J. Kampas, J.D. Pintér, Solving circle packing problems by global optimization: numerical results and industrial applications, Eur J Oper Res 191 (2008) 786–802.

[11] Z. Lü, W. Huang, PERM for solving circle packing problem, Comput Oper Res 35 (2008) 1742–1755.

[12] P. Amore, Circle packing in regular polygons, Physics of Fluids 35 (2023).

[13] R. Peikert, D. Würtz, M. Monagan, C. de Groot, Packing circles in a square: a review and new results, in: System Modelling and Optimization: Proceedings of the 15th IFIP Conference Zurich, Switzerland, September 2–6, 1991, Springer, 1992: pp. 45–54.

[14] J. Voglar, A. Peperko, Calculating the sequences behind a hexagonal lattice based equal circle packing in the Euclidian plane, ArXiv Preprint ArXiv:2403.10530 (2024).

[15] I. Castillo, F.J. Kampas, J.D. Pintér, Solving circle packing problems by global optimization: numerical results and industrial applications, Eur J Oper Res 191 (2008) 786–802.
[3] E.G. Birgin, J.M. Gentil, New and improved results for packing identical unitary radius circles within triangles, rectangles and strips, Comput Oper Res 37 (2010) 1318–1327.

[4] E. Specht, High density packings of equal circles in rectangles with variable aspect ratio, Comput Oper Res 40 (2013) 58–69.

[5] K.J. Nurmela, P.R.J. Östergård, Packing up to 50 equal circles in a square, Discrete Comput Geom 18 (1997) 111–120.

[6] P.G. Szabó, E. Specht, Packing up to 200 equal circles in a square, in: Models and Algorithms for Global Optimization: Essays Dedicated to Antanas Žilinskas on the Occasion of His 60th Birthday, Springer, 2007: pp. 141–156.

[7] P. Amore, D. de la Cruz, V. Hernandez, I. Rincon, U. Zarate, Circle packing in arbitrary domains, Physics of Fluids 35 (2023).

[8] C.O. López, J.E. Beasley, A heuristic for the circle packing problem with a variety of containers, Eur J Oper Res 214 (2011) 512–525.

[9] G.L. Orick, K. Stephenson, C. Collins, A linearized circle packing algorithm, Computational Geometry 64 (2017) 13–29.

[10] I. Castillo, F.J. Kampas, J.D. Pintér, Solving circle packing problems by global optimization: numerical results and industrial applications, Eur J Oper Res 191 (2008) 786–802.

[11] Z. Lü, W. Huang, PERM for solving circle packing problem, Comput Oper Res 35 (2008) 1742–1755.

[12] P. Amore, Circle packing in regular polygons, Physics of Fluids 35 (2023).

[13] R. Peikert, D. Würtz, M. Monagan, C. de Groot, Packing circles in a square: a review and new results, in: System Modelling and Optimization: Proceedings of the 15th IFIP Conference Zurich, Switzerland, September 2–6, 1991, Springer, 1992: pp. 45–54.

[14] J. Voglar, A. Peperko, Calculating the sequences behind a hexagonal lattice based equal circle packing in the Euclidian plane, ArXiv Preprint ArXiv:2403.10530 (2024).

[15] I. Castillo, F.J. Kampas, J.D. Pintér, Solving circle packing problems by global optimization: numerical results and industrial applications, Eur J Oper Res 191 (2008) 786–802.