Variations on the theme of Fibonacci

Variations on the given theme are a genre well known in music. The famous composer Mozart was a great amateur of this genre: for example, the first part of the famous Mozart's sonata A-dur is written in the form of the theme with variations. The first part of Beethoven's sonata As-dur also consists of variations on one theme. The distinctive feature of musical works of the variation genre consists of the fact, that they in the most cases begin with one simple essential theme, which hereinafter undergoes considerable changes on tempo, mood and nature. But how are the variation no bizarre, the listeners absolutely should have an impression that each of them is a natural development of the essential theme.

Let's follow to the example of musical composition and by selecting the simple mathematical subject (the series of Fibonacci numbers) we will consider it together with its numerous variations.

Lucas Numbers

Fibonacci had not become to study mathematical properties of the numerical series obtained by him:

 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … . (1)

This after him was made by other mathematicians. Since the 19th century the mathematical works dedicated to Fibonacci numbers, according to the witty expression of one mathematician, "began to reproduce as Fibonacci's rabbits".

The French mathematician Eduardo Lucas becames a leader of these researches in the 19th century. Lucas merit for Fibonacci numbers theory consists of the fact that he was first mathematician who introduced the name of "Fibonacci numbers" and besides who introduced into consideration so-called generalized Fibonacci numbers described with the following recurrent formula:

 Gn = Gn-1 + Gn-2. (2)

Depending on the initial terms G1 , G2 the recurrent formula (2) generates an infinite number of numerical series similar to Fibonacci numbers (1).

Among all possible series generated by (2) the greatest application had two numerical series, the Fibonacci numbers (1) and so-called Lucas numbers Ln, given with the following recurrent formula:

 Ln = Ln-1 + Ln-2. (3)

with the following initial terms:

 L1 = 1 è L2 = 2. (4)

Then, by using the recurrent formula (3) and the initial terms (4), we can compute the numerical series called as Lucas numbers:

 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, ... . (5)

The widened Fibonacci and Lucas numbers

Until now we considered Fibonacci (Fn) and Lucas (Ln) numbers implying that their indexes n are positive numbers, that is n = 0, 1, 2, 3, ... . It appears, that they can be widened to the side of the negative values of indexes n, that is, when the indexes n take their values from the set: n = -1, -2, -3, ... .

Fibonacci and Lucas numbers widened in this manner are presented in Table 1.

Table 1.

 n 0 1 2 3 4 5 6 7 8 9 10 Fn 0 1 1 2 3 5 8 13 21 34 55 F-n 0 1 -1 2 -3 5 -8 13 -21 34 -55 Ln 2 1 3 4 7 11 18 29 47 76 123 L-n 2 -1 3 -4 7 -11 18 -29 47 -76 123

As follows from Tab. 1 the terms of the widened series Fn and Ln have a number of wonderful mathematical properties. For example, for odd n = 2k + 1 the terms of the sequences Fn and F-n coincide, that is F2k+1 = F-2k-1, and for the even n = 2k they are opposite by the sign, that is: F2k = -F-2k. As to the Lucas numbers Ln, here all is contrary, that is: L2k = L-2k; L2k+1 = -L-2k-1.

And now we will consider closely the numerical series of Fibonacci and Lucas numbers given with Tab. 1. Let's consider, for example, Lucas number L4 = 7 and compare it with the series of Fibonacci numbers Fn. It is easy to see that L4 = 7 = 2 + 5. But numbers 2 and 5 are Fibonacci numbers F3 = 2 and F5 = 5.

But possibly is our observation an accidental coincidence? Continuing our investigation of Table 1, we will get the following: 1 = 0 + 1, 3 = 1 + 2, 4 = 1 + 3, 7 = 2 + 5, 11 = 3 + 8, 18 = 5 + 13, 29 = 8 + 21 etc. Let's compare now numerical series L-n and F-n. Here we get the same result, that is: -1 = 0 + (-1), 3 = 1 + 2, -4 = (-1) + (-3) etc. Thus, we have established the following surprised simple mathematical rule connecting Fibonacci and Lucas numbers:

Ln = Fn-1 + Fn+1,

where the index n takes the values from the set: 0, ±1, ±2, ±3, ... .

Continuing investigations of Table 1, it is possible also to establish that the Fibonacci and Lucas numbers are connected by other rather interesting identities, for example:

Ln = Fn + 2Fn-1; Ln + Fn = 2Fn+1 è ò.ä.

Fundamental identity connecting three adjacent Fibonacci numbers

Let's consider Fibonacci series: 1, 1, 2, 3, 5, 8, 13, 21, 34, …. Let's take Fibonacci number 5 and its square, that is: 52 = 25. Now we take the product of two adjacent Fibonacci numbers 3 and 8 encircled the number of 5, that is, 3 ´ 8 = 24. Then we can record:

52 - 3 ´ 8 = 1.

And now we do the same with the next Fibonacci number 8, that is at first we square it (82= 64), after that we calculate the product of two adjacent to 8 Fibonacci numbers of 5 and 13 (5 ´ 13 = 65) and then we subtract number 65 from number 82= 64:

82 - 5 ´ 13 = -1.

Note that the obtained difference is equal to (-1).

Further we have:

132 - 8 ´ 21 = 1,
212 - 13 ´ 34 = -1 è ò.ä.

We see, that the square of some Fibonacci number Fn always differs from the product of two adjacent Fibonacci numbers Fn-1 and Fn+1 encircled it by 1 and the sign of this 1 depends on the index n of the Fibonacci number Fn. If the index n is even then the number of 1 undertakes with minus, and if odd, with plus. The indicated property of Fibonacci numbers can be expressed by the following mathematical formula:

 (6)

This wonderful formula evokes a reverent thrill if to imagine that this one is valid for any value of n (we remind that n can be some integer in limits since -¥ up to +¥)and gives genuine aesthetic enjoying because the alternation of + 1 and -1 in the expression (6) at successive oversight of all Fibonacci numbers produces no realized feeling of a rhythm and harmony.