### Abstract

The metallic means (also known as metallic ratios) may be defined as the limiting ratio of consecutives terms of sequences connected to the Fibonacci sequence via the invert transform. For example, the Pell sequence (invert transform of the Fibonacci sequence) gives the so-called silver mean, and the invert transform of the Pell sequence leads to the bronze mean. The limiting ratio of the Fibonacci sequence itself is known as the golden mean or ratio. We introduce a new family of kth degree metallic means obtained through invert transforms of the generalized kth order Fibonacci sequence. As in the case k = 2, each generalized metallic mean is shown to be the unique positive root of a kth degree polynomial determined by the sequence.

Original language | English (US) |
---|---|

Pages (from-to) | 45-50 |

Number of pages | 6 |

Journal | Fibonacci Quarterly |

Volume | 57 |

Issue number | 1 |

State | Published - Feb 1 2019 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

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*Fibonacci Quarterly*, vol. 57, no. 1, pp. 45-50.

**Generalized metallic means.** / Gil, Juan Bautista; Worley, Aaron.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Generalized metallic means

AU - Gil, Juan Bautista

AU - Worley, Aaron

PY - 2019/2/1

Y1 - 2019/2/1

N2 - The metallic means (also known as metallic ratios) may be defined as the limiting ratio of consecutives terms of sequences connected to the Fibonacci sequence via the invert transform. For example, the Pell sequence (invert transform of the Fibonacci sequence) gives the so-called silver mean, and the invert transform of the Pell sequence leads to the bronze mean. The limiting ratio of the Fibonacci sequence itself is known as the golden mean or ratio. We introduce a new family of kth degree metallic means obtained through invert transforms of the generalized kth order Fibonacci sequence. As in the case k = 2, each generalized metallic mean is shown to be the unique positive root of a kth degree polynomial determined by the sequence.

AB - The metallic means (also known as metallic ratios) may be defined as the limiting ratio of consecutives terms of sequences connected to the Fibonacci sequence via the invert transform. For example, the Pell sequence (invert transform of the Fibonacci sequence) gives the so-called silver mean, and the invert transform of the Pell sequence leads to the bronze mean. The limiting ratio of the Fibonacci sequence itself is known as the golden mean or ratio. We introduce a new family of kth degree metallic means obtained through invert transforms of the generalized kth order Fibonacci sequence. As in the case k = 2, each generalized metallic mean is shown to be the unique positive root of a kth degree polynomial determined by the sequence.

UR - http://www.scopus.com/inward/record.url?scp=85063084509&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85063084509&partnerID=8YFLogxK

M3 - Article

VL - 57

SP - 45

EP - 50

JO - Fibonacci Quarterly

JF - Fibonacci Quarterly

SN - 0015-0517

IS - 1

ER -