# Efficient Location of Resources in Cylindrical Networks

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Dominating a Cylinder as Efficiently as Possible

**Proposition**

**1.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Remark**

**1.**

**Proposition**

**4.**

**Proof.**

**Remark**

**2.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 3. Computing the Independent [$\mathbf{1},\mathbf{2}$]-Number in Cylinders

- $u=0$ if $u\in S$;
- $u=1$ if u has exactly one neighbor in S in its column or in the previous one;
- $u=2$ if u has exactly two neighbors in S in its column or in the previous one;
- $u=3$ if u has no neighbors in S in its column or in the previous one.

- (a)
- $00,22,33,112,211,212,213,321,1111,1113,3111,3113$
- (b)
- $03,30,010,121,123,321,323$
- (c)
- $12021,12023,32021,32023$

- (i)
- if ${q}_{i}=0$, then either $\left\{{p}_{i}=2\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\{{p}_{i-1}\ne 0\phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{4.pt}{0ex}}{p}_{i+1}\ne 0,\phantom{\rule{4.pt}{0ex}}\mathrm{but}\phantom{\rule{4.pt}{0ex}}\mathrm{not}\phantom{\rule{4.pt}{0ex}}\mathrm{both}\phantom{\rule{4.pt}{0ex}}\mathrm{are}\phantom{\rule{4.pt}{0ex}}\mathrm{non}-\mathrm{zero}\}\right\}$ or $\left\{{p}_{i}=1\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}{p}_{i-1}\ne 0\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}{p}_{i+1}\ne 0\right\}$;
- (ii)
- if ${q}_{i}=1$, then either ${p}_{i}=0$ or $\left\{{p}_{i}=1\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\{{p}_{i-1}\ne 0\phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{4.pt}{0ex}}{p}_{i+1}\ne 0,\phantom{\rule{4.pt}{0ex}}\mathrm{but}\phantom{\rule{4.pt}{0ex}}\mathrm{not}\phantom{\rule{4.pt}{0ex}}\mathrm{both}\phantom{\rule{4.pt}{0ex}}\mathrm{are}\phantom{\rule{4.pt}{0ex}}\mathrm{non}-\mathrm{zero}\}\right\}$ or $\left\{{p}_{i}=2\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}{p}_{i-1}=0\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}{p}_{i+1}=0\right\}$ or ${p}_{i}=3$;
- (iii)
- if ${q}_{i}=2$, then either $\left\{{p}_{i}=1\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\{{p}_{i-1}\ne 0\phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{4.pt}{0ex}}{p}_{i+1}\ne 0,\phantom{\rule{4.pt}{0ex}}\mathrm{but}\phantom{\rule{4.pt}{0ex}}\mathrm{not}\phantom{\rule{4.pt}{0ex}}\mathrm{both}\phantom{\rule{4.pt}{0ex}}\mathrm{are}\phantom{\rule{4.pt}{0ex}}\mathrm{non}-\mathrm{zero}\}\right\}$ or ${p}_{i}=3$;
- (iv)
- if ${q}_{i}=3$, then ${p}_{i}=0$.

- ${P}_{1}$ is initial;
- ${P}_{i+1}$ can follow ${P}_{i}$, for $i\in \{1,2,\cdots n-1\}$;
- ${P}_{n}$ is final.

**Proposition**

**5.**

**Proof.**

Algorithm 1: Recurrence for the Independent $[1,2]$-number in Cylinders |

## 4. Experimental Results

#### 4.1. Cases $m=$ 4 and $m=$ 5

#### 4.2. Cases 6 $\le m\le $ 10

- both are infinite;
- both are finite, then we compute the difference ${X}^{20+d}\left(i\right)-{X}^{20}\left(i\right)$;
- one of them is finite and the other one is infinite, then we say that they are non-comparable.

- ${i}_{[1,2]}({C}_{6}\square {P}_{n})=\left\{\begin{array}{cc}6\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}n=3,\hfill \\ 9\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}n=5,\hfill \\ \u2308\frac{4n}{3}\u2309\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}n\equiv 1\phantom{\rule{4.44443pt}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}3\right),\hfill \\ \u2308\frac{4n}{3}\u2309+1\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{otherwise}.\hfill \end{array}\right.$
- ${i}_{[1,2]}({C}_{7}\square {P}_{n})=\left\{\begin{array}{cc}2n\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}n=2,3,4,\hfill \\ \u2308\frac{3n}{2}\u2309+2\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}n\equiv 1\phantom{\rule{4.44443pt}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}2\right),\hfill \\ \u2308\frac{3n}{2}\u2309+3\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{otherwise}.\hfill \end{array}\right.$
- ${i}_{[1,2]}({C}_{8}\square {P}_{n})=\left\{\begin{array}{cc}2n\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}n=2,3,4,5,7,9,\hfill \\ \u2308\frac{9n}{5}\u2309+1\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}n\equiv 1,3\phantom{\rule{4.44443pt}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}5\right),\hfill \\ \u2308\frac{9n}{5}\u2309+2\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}n\equiv 0\phantom{\rule{4.44443pt}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}5\right)\phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{4.pt}{0ex}}n\le 100,n\equiv 2,4\phantom{\rule{4.44443pt}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}5\right).\hfill \end{array}\right.$
- $\u2308\frac{9n}{5}\u2309+1\le {i}_{[1,2]}({C}_{8}\square {P}_{n})\le \u2308\frac{9n}{5}\u2309+2$, if $n>100$ and $n\equiv 2,4\phantom{\rule{4.44443pt}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}5\right).$
- ${i}_{[1,2]}({C}_{9}\square {P}_{n})=2n+2$, if $n\ge 2$.
- ${i}_{[1,2]}({C}_{10}\square {P}_{n})=\left\{\begin{array}{cc}2n+2\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}n=2,4,5,\hfill \\ 2n+3\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}n=3,6,7,8,9,\hfill \\ 2n+4\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{otherwise}\phantom{\rule{4.pt}{0ex}}.\hfill \end{array}\right.$

#### 4.3. Cases 11 $\le m\le $ 15

- $u=0$ if $u\in S$;
- $u=1$ if u has at least one neighbor in S, in its column or in the previous one;
- $u=2$ if u has no neighbors in S, in its column or in the previous one.

- (i)
- if ${q}_{i}=0$, then ${p}_{i}=1$,
- (ii)
- if ${q}_{i}=1$, then either ${p}_{i}=0$ or $\left\{{p}_{i}=1\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\{{p}_{i-1}\ne 0\phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{4.pt}{0ex}}{p}_{i+1}\ne 0,\}\right\}$ or ${p}_{i}=2$,
- (iii)
- if ${q}_{i}=2$, then ${p}_{i}=0$.

- ${i}_{[1,2]}({C}_{11}\square {P}_{n})=\left\{\begin{array}{cc}6\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}n=2,\hfill \\ 9\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}n=3,\hfill \\ \u2308\frac{12n+12}{5}\u2309\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{otherwise}.\hfill \end{array}\right.$
- ${i}_{[1,2]}({C}_{12}\square {P}_{n})=\left\{\begin{array}{cc}3n\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}n=2,3,4,\hfill \\ \u2308\frac{5n+5}{2}\u2309\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}n=5,6,7,\hfill \\ \u2308\frac{5n+9}{2}\u2309\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}n\ge 8.\hfill \end{array}\right.$
- ${i}_{[1,2]}({C}_{13}\square {P}_{n})=\left\{\begin{array}{cc}7\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}n=2,\hfill \\ 10\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}n=3,\hfill \\ \u2308\frac{20n+11}{7}\u2309\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}4\le n\le 9,\hfill \\ \u2308\frac{20n+20}{7}\u2309\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}n\ge 10,n\neg \equiv 0,2\phantom{\rule{4.44443pt}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}7\right)\phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{4.pt}{0ex}}n=14,16,\hfill \\ \u2308\frac{20n+20}{7}\u2309+1\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}21\le n\le 100,n\equiv 0,2\phantom{\rule{4.44443pt}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}7\right).\hfill \end{array}\right.$
- $\u2308\frac{20n+20}{7}\u2309\le {i}_{[1,2]}({C}_{13}\square {P}_{n})\le \u2308\frac{20n+20}{7}\u2309+1$, if $n>100$ and $n\equiv 0,2\phantom{\rule{4.44443pt}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}7\right).$
- ${i}_{[1,2]}({C}_{14}\square {P}_{n})=\left\{\begin{array}{cc}8\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}n=2,\hfill \\ 3n+3\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}3\le n\le 9,\hfill \\ 3n+4\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{otherwise}.\hfill \end{array}\right.$
- ${i}_{[1,2]}({C}_{15}\square {P}_{n})=\left\{\begin{array}{cc}8\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}n=2,\hfill \\ 12\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}n=3,\hfill \\ 3n+4\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}n=4,5,\hfill \\ 3n+5\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}n=6,7,\hfill \\ 3n+6\hfill & \phantom{\rule{4.pt}{0ex}}\mathrm{otherwise}.\hfill \end{array}\right.$

## 5. Conclusions

- if $m=3,4,5,6,9,10,15$, then ${i}_{[1,2]}({C}_{m}\square {P}_{n})=\gamma ({C}_{m}\square {P}_{n})$;
- if $m=7,14$, then ${i}_{[1,2]}({C}_{m}\square {P}_{n})=\gamma ({C}_{m}\square {P}_{n})+1$;
- if $m=8,12$, then $\gamma ({C}_{m}\square {P}_{n})\le {i}_{[1,2]}({C}_{m}\square {P}_{n})\le \gamma ({C}_{m}\square {P}_{n})+1$,
- ${i}_{[1,2]}({C}_{11}\square {P}_{n})=i({C}_{11}\square {P}_{n})$;
- $i({C}_{13}\square {P}_{n})\le {i}_{[1,2]}({C}_{13}\square {P}_{n})\le i({C}_{13}\square {P}_{n})+1$.

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Regular patterns for an independent $[1,2]$-set (black vertices) in ${C}_{m}\square {P}_{2}$, $m\ne 5$: (

**a**) $m\equiv 0\phantom{\rule{4.44443pt}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}4\right),\phantom{\rule{4pt}{0ex}}m\ge 4$; (

**b**) $m=9$; (

**c**) $m\equiv 1\phantom{\rule{4.44443pt}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}4\right),\phantom{\rule{4pt}{0ex}}m\ge 13$; (

**d**) $m=6$; (

**e**) $m\equiv 2\phantom{\rule{4.44443pt}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}4\right),\phantom{\rule{4pt}{0ex}}m\ge 10$; (

**f**) $m=3$; (

**g**) $m\equiv 3\phantom{\rule{4.44443pt}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}4\right),\phantom{\rule{4pt}{0ex}}m\ge 7$.

**Figure 4.**$\gamma ({C}_{3}\square {P}_{n})<i({C}_{3}\square {P}_{n})$, for every $n\ge 5$: (

**a**) $\gamma ({C}_{3}\square {P}_{8})=7$; (

**b**) $\gamma ({C}_{3}\square {P}_{9})=7$.

**Figure 5.**Regular pattern (black vertices) for an independent $[1,2]$-set in ${C}_{2r}\square {P}_{2k+1}$.

**Figure 6.**Regular pattern (black vertices) for an independent $[1,2]$-set in ${C}_{2r+1}\square {P}_{4k+1}$.

**Figure 7.**Regular pattern (black vertices) for an independent $[1,2]$-set in ${C}_{2r+1}\square {P}_{4k+3}$.

**Figure 8.**Regular pattern (black vertices) for an independent $[1,2]$-set in ${C}_{3r}\square {P}_{n}$.

**Figure 9.**Construction of a regular pattern for an independent $[1,2]$-set in ${C}_{3r+1}\square {P}_{2k}$: (

**a**) first column; (

**b**) a pair of columns; (

**c**) two pairs of columns; (

**d**) k pairs of columns.

**Figure 10.**Construction of a regular pattern for an independent $[1,2]$-set in ${C}_{3r+2}\square {P}_{2k}$: (

**a**) Type A column; (

**b**) Type B column; (

**c**) four-column block; (

**d**) two-column block; (

**e**), (

**f**) joining blocks.

**Figure 11.**Regular pattern (black vertices) for an independent $[1,2]$-set in ${C}_{3r+2}\square {P}_{2k}$.

m | ${4}^{\mathit{m}}=$ Words of Length m in Alphabet $\{0,1,2,3\}$ | ${\mathit{c}}_{\mathit{m}}=$ Number of Correct Words | Execution Times | ||
---|---|---|---|---|---|

Correct Words | A | ${\mathit{X}}^{\mathit{r}}$, $2\le \mathit{r}\le 100$ | |||

4 | 256 | 20 | $<1$ s. | $<1$ s. | $<1$ s. |

5 | 1024 | 35 | $<1$ s. | $<1$ s. | $<1$ s. |

6 | 4096 | 79 | $<1$ s. | $<1$ s. | $<1$ s. |

7 | 16384 | 154 | $<1$ s. | $<1$ s. | $<1$ s. |

8 | 65536 | 332 | $<1$ s. | $<1$ s. | $<1$ s. |

9 | 262144 | 666 | $<1$ s. | $<1$ s. | $<1$ s. |

10 | 1048576 | 1389 | $<1$ s. | $<1$ s. | $<1$ s. |

11 | 4194304 | 2849 | $<1$ s. | $2.6$ s. | $<1$ s. |

12 | 16777216 | 5891 | $2.2$ s. | $11.8$ s. | $1.9$ s. |

13 | 67108864 | 12116 | $8.1$ s. | $53.2$ s. | $3.7$ s. |

14 | 268435456 | 25008 | $31.1$ s. | 4 m. 3 s. | $6.9$ s. |

15 | 1073741824 | 51509 | 2 m. 5 s. | 18 m. 25 s. | $13.3$ s. |

m | ${\mathit{n}}_{0}$ | d | c | Finite Difference Equation | Boundary Values | Rest of Values |
---|---|---|---|---|---|---|

4 | 3 | 2 | 2 | ${i}_{[1,2]}({C}_{4}\square {P}_{n+2})-{i}_{[1,2]}({C}_{4}\square {P}_{n})=2,n\ge 3$ | ${i}_{[1,2]}({C}_{4}\square {P}_{3})=3$ ${i}_{[1,2]}({C}_{4}\square {P}_{4})=4$ | ${i}_{[1,2]}({C}_{4}\square {P}_{2})=2$ |

5 | 4 | 1 | 1 | ${i}_{[1,2]}({C}_{5}\square {P}_{n+1})-{i}_{[1,2]}({C}_{5}\square {P}_{n})=1,n\ge 4$ | ${i}_{[1,2]}({C}_{5}\square {P}_{4})=6$ | ${i}_{[1,2]}({C}_{5}\square {P}_{3})=5$ |

m | d | c | Vector Pair ${\mathit{X}}^{20+\mathit{d}},{\mathit{X}}^{20}$ | Values of Differences | Non-Comparable Pairs | Remove Correct Words in Positions Where Appears | Remaining Words |
---|---|---|---|---|---|---|---|

6 | 3 | 4 | ${X}^{23},{X}^{20}$ | $4,6$ | yes | 6, non-comparable | 69 |

7 | 2 | 3 | ${X}^{22},{X}^{20}$ | $3,4$ | no | 4 | 126 |

8 | 5 | 9 | ${X}^{25},{X}^{20}$ | $9,10$ | yes | 10, non-comparable | 228 |

9 | 2 | 4 | ${X}^{22},{X}^{20}$ | $4,6$ | no | 6 | 660 |

10-I | 2 | 4 | ${X}^{22},{X}^{20}$ | $2,4,5,6$ | no | $2,5,6$ | 1077 |

10-II | 2 | 4 | ${X}^{22},{X}^{20}$ | $4,5$ | no | 5 | 1067 |

m | ${\mathit{n}}_{0}$ | d | c | Auxiliary Equation | Boundary Values |
---|---|---|---|---|---|

6 | 7 | 3 | 4 | ${f}_{6}(n+3)-{f}_{6}\left(n\right)=4,n\ge 7$ | ${f}_{6}\left(7\right)=10,{f}_{6}\left(8\right)=12,{f}_{6}\left(9\right)=13$ |

7 | 6 | 2 | 3 | ${f}_{7}(n+2)-{f}_{7}\left(n\right)=3,n\ge 6$ | ${f}_{7}\left(6\right)=12,{f}_{7}\left(7\right)=13$ |

8 | 11 | 5 | 9 | ${f}_{8}(n+5)-{f}_{8}\left(n\right)=9,n\ge 11$ | ${f}_{8}\left(11\right)=21,{f}_{8}\left(12\right)=24,{f}_{8}\left(13\right)=25$ ${f}_{8}\left(14\right)=28,{f}_{8}\left(15\right)=29$ |

9 | 8 | 2 | 4 | ${f}_{9}(n+2)-{f}_{9}\left(n\right)=4,n\ge 8$ | ${f}_{9}\left(8\right)=18,{f}_{9}\left(9\right)=20$ |

10 | 10 | 2 | 4 | ${f}_{10}(n+2)-{f}_{10}\left(n\right)=4,n\ge 10$ | ${f}_{10}\left(10\right)=24,{f}_{10}\left(11\right)=26$ |

m | d | c | Vector Pair ${\mathit{X}}^{50+\mathit{d}},{\mathit{X}}^{50}$ | Values of Differences | Non-Comparable Pairs | Remove Correct Words in Positions Where Appears | Remaining Words |
---|---|---|---|---|---|---|---|

11 | 5 | 12 | ${X}^{55},{X}^{50}$ | $11,12,13,14,15$ | no | $11,13,14,15$ | 2475 |

12 | 2 | 5 | ${X}^{52},{X}^{50}$ | $2,4,5,6,8$ | no | $2,4,6,8$ | 3531 |

13 | 7 | 20 | ${X}^{57},{X}^{50}$ | $19,20,21,22,23,28$ | no | $19,21,22,23,28$ | 9438 |

14-I | 2 | 6 | ${X}^{52},{X}^{50}$ | $2,4,5,6,7,8$ | no | $2,4,5,7,8$ | 20792 |

14-II | 2 | 6 | ${X}^{52},{X}^{50}$ | $6,7$ | no | 7 | 19686 |

15 | 1 | 3 | ${X}^{51},{X}^{50}$ | $1,2,3,4,5,6$ | no | $1,2,4,5,6$ | 34913 |

m | ${\mathit{n}}_{0}$ | d | c | Auxiliary Equation | Boundary Values |
---|---|---|---|---|---|

11 | 7 | 5 | 12 | ${f}_{11}(n+5)-{f}_{11}\left(n\right)=12,n\ge 7$ | ${f}_{11}\left(7\right)=20,{f}_{11}\left(8\right)=22,{f}_{11}\left(9\right)=24$, ${f}_{11}\left(10\right)=27,{f}_{11}\left(11\right)=29$ |

12 | 8 | 2 | 5 | ${f}_{12}(n+2)-{f}_{12}\left(n\right)=5,n\ge 8$ | ${f}_{12}\left(8\right)=25,{f}_{12}\left(9\right)=27$ |

13 | 10 | 7 | 20 | ${f}_{13}(n+7)-{f}_{13}\left(n\right)=20,n\ge 10$ | ${f}_{13}\left(10\right)=32,{f}_{13}\left(11\right)=35,{f}_{13}\left(12\right)=38$, ${f}_{13}\left(13\right)=40,{f}_{13}\left(14\right)=44,{f}_{13}\left(15\right)=46,{f}_{13}\left(16\right)=50$ |

14 | 10 | 2 | 6 | ${f}_{14}(n+2)-{f}_{14}\left(n\right)=6,n\ge 10$ | ${f}_{14}\left(10\right)=34,{f}_{14}\left(11\right)=37$ |

15 | 10 | 1 | 3 | ${f}_{15}(n+1)-{f}_{15}\left(n\right)=3,n\ge 10$ | ${f}_{15}\left(10\right)=36$ |

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**MDPI and ACS Style**

Carreño, J.J.; Martínez, J.A.; Puertas, M.L.
Efficient Location of Resources in Cylindrical Networks. *Symmetry* **2018**, *10*, 24.
https://doi.org/10.3390/sym10010024

**AMA Style**

Carreño JJ, Martínez JA, Puertas ML.
Efficient Location of Resources in Cylindrical Networks. *Symmetry*. 2018; 10(1):24.
https://doi.org/10.3390/sym10010024

**Chicago/Turabian Style**

Carreño, José Juan, José Antonio Martínez, and María Luz Puertas.
2018. "Efficient Location of Resources in Cylindrical Networks" *Symmetry* 10, no. 1: 24.
https://doi.org/10.3390/sym10010024