Reverse Fibonacci sequence and its description

Jakub Souček1, Ondrej Janíčko2

1   Pilsen, Czech Republic,

2   Bratislava, Slovak Republic,,

19.1. 2019

Abstract: This article describes a new sequence called „Reverse Fibonacci sequence“. The introduction of the study deals with the derivation of limits of a ratio of the two following numbers of the proposed sequence, which is equal to the number j. Further, the individual properties of this number are described. Another part of the article is devoted to the geometric interpretationof the obtained number j. With the help of the conclusions of the geometrical interpretation, the individual relations of the number j to the number  π , the golden ratio φ, the Euler number e and spiral of Theodorus are derived. The conclusion of the article deals with the relationship of the proposed sequence to the Fibonacci sequence. The proposed sequence is characterized by the same features as the Fibonacci sequence for individual values of the digital root of members of its sequence. The values of digital roots of the proposed sequence are reversed to the Fibonacci sequence values.

Keywords: Reverse Fibonacci sequence, Fibonacci sequence, Lucas sequence, digital root, number π, Euler number e, golden ratio j, spiral of Theodorus, Vesicapiscis

1. Introduction
2. Reverse Fibonacci sequence
3. Geometric interpretation
3.1 Relation of j to the number π
3.2 Relation of j to the golden ratio φ
3.3 Relation of j to Euler’s number e
3.4 Relation of j to the spiral of Theodorus
4. Relation of proposed sequence with Fibonacci sequence

5. Conclusion
The presented article defines a new sequence and a new ratio of the so-called reverse Fibonacci sequence and their relations and properties to other mathematical fundamental numbers and structures, which points to the interest of this sequence and the derived ratio. A definite relation between the original Fibonacci sequence and the proposed reverse Fibonacci sequence and their ratios is proved. The Fibonacci sequence and the golden ratio are applied in nature; therefore, we can assume that in the future the application for the reverse Fibonacci sequence will be discovered in nature. The reverse Fibonacci sequence is hiden in the Fibonacci sequence, therefore all the newly derived properties of the reverse Fibonacci sequence can be used in further golden ratio research and a self-directed field dedicated to the Fibonacci sequence. Also very important is the derived relationship between the golden ratio and the ratio of the reverse Fibonacci sequence. Another important relationship is between the ratio of the reverse Fibonacci sequence and Euler’s numbere.This article expands our knowledge of the golden ratio and the Fibonacci sequence, and therefore gives hope for widespread use in various disciplines, art and nature.

Link to full article is here  Reverse_Fibonacci_Sequence_Soucek_Janicko.pdf