Generalizations of Fibonacci numbers

In mathematics, the Fibonacci numbers form a sequence defined recursively by a special rule: after two starting values, each number is the sum of the two numbers before it. This simple idea creates a sequence that appears in many areas of nature and mathematics.

The Fibonacci sequence has been studied a lot, and mathematicians have found many ways to generalize it. They explore sequences that start with different numbers instead of 0 and 1. They also look at sequences where each new number is the sum of more than two previous numbers. These generalizations help us understand patterns in nature and solve many types of problems.

Extension to negative integers

We can extend the Fibonacci sequence to include negative numbers. By using a special rule, we can find Fibonacci numbers for negative positions. This gives us a sequence that includes both positive and negative numbers, like ..., −8, 5, −3, 2, −1, 1, 0, 1, 1, 2, 3, 5, 8, ...

See also Negafibonacci coding.

Extension to all real or complex numbers

The Fibonacci sequence can be extended to include real and complex numbers using special formulas. These extensions involve the golden ratio, a special number often denoted by φ. One such formula, known as Binet's formula, helps calculate Fibonacci numbers using powers of the golden ratio.

Mathematicians have created functions that work for any real or complex number, not just whole numbers. These functions follow the same adding rule as the Fibonacci sequence but can be used to find values for numbers like 3 + 4i. There are many ways to extend the Fibonacci sequence to complex numbers, showing how flexible and interesting this sequence can be.

Vector space

The Fibonacci sequence can also be thought of in more abstract ways. We can apply the same idea of adding the last two numbers to get the next one, but instead of using regular numbers, we can use different kinds of mathematical objects.

When we do this with certain types of numbers or structures, the set of all such sequences forms something called a vector space. This means they follow special rules that make them useful in more advanced mathematics.

Similar integer sequences

Integer sequences can follow rules similar to the Fibonacci sequence, where each number is the sum of the previous ones. For example, the Fibonacci sequence starts with 0 and 1, and each new number is the sum of the two before it. This idea can be expanded in many ways.

One common expansion is to start with different numbers or to add more than two numbers to find the next one. These variations create new sequences with unique patterns and properties, often studied in number theory.

Fibonacci word

Main article: Fibonacci word

Just like the numbers in the Fibonacci sequence, there is a special set of strings called the Fibonacci word. These strings are made by joining smaller strings together, following a pattern. The first few strings are: b, a, ab, aba, abaab, and so on. Each string’s length matches a number in the Fibonacci sequence. These strings are important in computer science because they can show the most challenging cases for some algorithms. They are also used in science to model special structures called Fibonacci quasicrystals, which have unique properties.

The process of building these strings is similar to the way we add numbers in the Fibonacci sequence, but instead of adding numbers, we join strings together.

Convolved Fibonacci sequences

A convolved Fibonacci sequence is made by doing a special math operation called convolution on the regular Fibonacci numbers many times. This creates new sequences of numbers.

Some of the first convolved sequences start like this:

• For r = 1: 0, 0, 1, 2, 5, 10, 20, 38, 71, …
• For r = 2: 0, 0, 0, 1, 3, 9, 22, 51, 111, …
• For r = 3: 0, 0, 0, 0, 1, 4, 14, 40, 105, …

These sequences can be calculated using special rules, and they are connected to other math ideas like Fibonacci polynomials. For example, one of these sequences counts how many ways to write a number using only 0, 1, and 2, with 0 used exactly once.

Other generalizations

The Fibonacci polynomials are another way to expand on Fibonacci numbers. Other sequences, like the Padovan sequence, follow different rules. For example, instead of adding the last two numbers, it adds the numbers two and three places back.

There are also special sequences like the random Fibonacci sequence, where a coin flip decides whether to add or subtract the previous numbers. Amazingly, this sequence grows in a predictable way, even with random choices.

A repfigit, or Keith number, is a number that appears in a Fibonacci-like sequence it starts. For example, 47 works because starting with 4 and 7, the sequence soon reaches 47. These numbers have unique patterns and properties that mathematicians find fascinating.