Generalizations of Fibonacci numbers

The Fibonacci numbers are a special list of numbers used in mathematics. The list follows a simple rule: after the first two numbers, each new number is the sum of the two numbers before it.

People have studied the Fibonacci numbers a lot. They have found many ways to make new lists like it. These new lists can start with different numbers. They can also use a rule where each new number is the sum of more than two numbers before it. These ideas help us see patterns in nature and solve different kinds 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 positions that are negative. 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 be thought of in more abstract ways. We can use the same idea of adding the last two numbers to get the next one, but with 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 like the Fibonacci sequence. The Fibonacci sequence starts with 0 and 1. Each new number is the sum of the two numbers before it. This idea can be changed in many ways.

One common change is to start with different numbers. Another is to add more than two numbers to find the next one. These changes make new sequences with special patterns. People often study these 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. 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 makes new lists 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 follow special rules. They are linked 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 build 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. This sequence still 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 special patterns that mathematicians find interesting.