# Almost Repdigit k-Fibonacci Numbers with an Application of k-Generalized Fibonacci Sequences

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## Abstract

**:**

## 1. Introduction

**Theorem 1.**

## 2. The Tools

- $h({\theta}_{1}\pm {\theta}_{2})\le h\left({\theta}_{1}\right)+h\left({\theta}_{2}\right)+log2$.
- $h\left({\theta}_{1}{\theta}_{2}^{\pm 1}\right)\le h\left({\theta}_{1}\right)+h\left({\theta}_{2}\right)$.
- $h\left({\theta}^{s}\right)=\left|s\right|h\left(\theta \right),$$s\in \mathbb{Z}$.

**Theorem 2**

**.**Assume that ${\alpha}_{1},\dots ,{\alpha}_{t}$ are positive real algebraic numbers in a real algebraic number field $\mathbb{K}$ of degree ${d}_{\mathbb{K}}$ and let ${b}_{1},\dots ,{b}_{t}$ be rational integers, such that

**Lemma 1**

**.**Let M be a positive integer, and let $p/q$ be a convergent of the continued fraction of the irrational τ such that $q>6M.$ Let $A,B,\mu $ be some real numbers with $A>0$ and $B>1$. If $\u03f5:=\left|\right|\mu q\left|\right|-M\left|\right|\tau q\left|\right|>0$, then there is no solution to the inequality

## 3. Properties of $\mathit{k}$-Fibonacci Numbers

## 4. Proof of Theorem 1

#### 4.1. The Case $n\le k+1$ and Almost Repdigits of the Form ${2}^{n}$

#### 4.2. A Bound for n Depending on k

**Lemma 2.**

#### 4.3. The Case $k\le 470$

#### 4.4. The Case $k>470$

**Lemma 3**

**.**If $n<{2}^{k/2},$ then the following estimates hold:

#### 4.5. Reducing the Bound on k

#### 4.6. The Case $a=0$ and k-Fibonacci Numbers as Powers of 10

## 5. An Application of $\mathit{k}$-Generalized Tiny Golden Angles to MR Imaging

## 6. Discussion

## 7. Recommendations

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Table A1.**The First 15 Elements of the Sequence ${F}_{n}^{\left(k\right)}:$ for $2\le k\le 9$ and $1\le n\le 15$. The circled numbers are all almost repdigits given in Theorem 1.

k = 2 | ${F}_{n}^{\left(2\right)}:$ | 1, | 1, | 2, | 3, | 5, | 8, | 13, | 21, | 34, | 55, | 89, | $\overline{)144}$, | $\overline{)233}$, | $\overline{)377}$, | 610 |

k = 3 | ${F}_{n}^{\left(3\right)}:$ | 1, | 1, | 2, | 4, | 7, | 13, | 24, | 44, | 81, | 149, | 274, | 504, | 927, | 1705, | 3136 |

k = 4 | ${F}_{n}^{\left(4\right)}:$ | 1, | 1, | 2, | 4, | 8, | 15, | 29, | 56, | 108, | 208, | 401, | $\overline{)773}$, | 1490, | 2872, | 5536 |

k = 5 | ${F}_{n}^{5)}:$ | 1, | 1, | 2, | 4, | 8, | 16, | 31, | 61, | 120, | 236, | $\overline{)464}$, | 912, | 1793, | 3525, | 6930 |

k = 6 | ${F}_{n}^{\left(6\right)}:$ | 1, | 1, | 2, | 4, | 8, | 16, | 32, | 63, | 125, | 248, | 492, | 976, | 1936, | 3840, | 7617 |

k = 7 | ${F}_{n}^{\left(7\right)}:$ | 1, | 1, | 2, | 4, | 8, | 16, | 32, | 64, | 127, | 253, | 504, | 1004, | $\overline{)2000}$, | 3984, | 7936 |

k = 8 | ${F}_{n}^{\left(8\right)}:$ | 1, | 1, | 2, | 4, | 8, | 16, | 32, | 64, | 128, | $\overline{)255}$, | 509, | 1016, | 2028, | 4048, | 8080 |

k = 9 | ${F}_{n}^{\left(9\right)}:$ | 1, | 1, | 2, | 4, | 8, | 16, | 32, | 64, | 128, | 256, | $\overline{)511}$, | 1021, | 2040, | 4076, | 8144 |

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**Table 1.**The First Ten Elements of the Sequence ${\psi}_{N}^{\left(k\right)}$ for $k=2,3,4,7$ and 10 as degree.

N | ${\mathit{\psi}}_{\mathit{N}}^{\left(2\right)}$ | ${\mathit{\psi}}_{\mathit{N}}^{\left(3\right)}$ | ${\mathit{\psi}}_{\mathit{N}}^{\left(4\right)}$ | ${\mathit{\psi}}_{\mathit{N}}^{\left(7\right)}$ | ${\mathit{\psi}}_{\mathit{N}}^{\left(10\right)}$ |
---|---|---|---|---|---|

1 | $111.24611$…${}^{\circ}$ | $116.60379..{.}^{\circ}$ | $118.51539..{.}^{\circ}$ | $119.83884..{.}^{\circ}$ | $119.98031..{.}^{\circ}$ |

2 | $68.75388..{.}^{\circ}$ | $70.76336..{.}^{\circ}$ | $71.46288..{.}^{\circ}$ | $71.94195..{.}^{\circ}$ | $71.99291..{.}^{\circ}$ |

3 | $49.75077..{.}^{\circ}$ | $50.79452..{.}^{\circ}$ | $51.15394..{.}^{\circ}$ | $51.39894..{.}^{\circ}$ | $51.42495..{.}^{\circ}$ |

4 | $38.97762..{.}^{\circ}$ | $39.61538..{.}^{\circ}$ | $39.83367..{.}^{\circ}$ | $39.98207..{.}^{\circ}$ | $39.99781..{.}^{\circ}$ |

5 | $32.03967..{.}^{\circ}$ | $32.46935..{.}^{\circ}$ | $32.61584..{.}^{\circ}$ | $32.71527..{.}^{\circ}$ | $32.72580..{.}^{\circ}$ |

6 | $27.19840..{.}^{\circ}$ | $27.50741..{.}^{\circ}$ | $27.61248..{.}^{\circ}$ | $27.68371..{.}^{\circ}$ | $27.69125..{.}^{\circ}$ |

7 | $23.62814..{.}^{\circ}$ | $23.86100..{.}^{\circ}$ | $23.94002..{.}^{\circ}$ | $23.99354..{.}^{\circ}$ | $23.99921..{.}^{\circ}$ |

8 | $20.88643..{.}^{\circ}$ | $21.06818..{.}^{\circ}$ | $21.12976..{.}^{\circ}$ | $21.17144..{.}^{\circ}$ | $21.17585..{.}^{\circ}$ |

9 | $18.71484..{.}^{\circ}$ | $18.86063..{.}^{\circ}$ | $18.90996..{.}^{\circ}$ | $18.94334..{.}^{\circ}$ | $18.94687..{.}^{\circ}$ |

10 | $16.95229..{.}^{\circ}$ | $17.07182..{.}^{\circ}$ | $17.11223..{.}^{\circ}$ | $17.13956..{.}^{\circ}$ | $17.14245..{.}^{\circ}$ |

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**MDPI and ACS Style**

Altassan, A.; Alan, M. Almost Repdigit *k*-Fibonacci Numbers with an Application of *k*-Generalized Fibonacci Sequences. *Mathematics* **2023**, *11*, 455.
https://doi.org/10.3390/math11020455

**AMA Style**

Altassan A, Alan M. Almost Repdigit *k*-Fibonacci Numbers with an Application of *k*-Generalized Fibonacci Sequences. *Mathematics*. 2023; 11(2):455.
https://doi.org/10.3390/math11020455

**Chicago/Turabian Style**

Altassan, Alaa, and Murat Alan. 2023. "Almost Repdigit *k*-Fibonacci Numbers with an Application of *k*-Generalized Fibonacci Sequences" *Mathematics* 11, no. 2: 455.
https://doi.org/10.3390/math11020455