Sequences and Series: Fibonacci and related numbers

Understanding Fibonacci Sequence

The Fibonacci sequence is a prominent example of a recurrence relation. It is defined as f(0) = 0, f(1) = 1, and f(n) = f(n-1) + f(n-2) for n > 1.
Each term in the Fibonacci sequence is the sum of the two preceding ones, starting from 0 and 1.
The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, also known as Fibonacci.

Fibonacci and Golden Ratio

The golden ratio, often symbolised by the Greek letter phi (Φ), is intimately associated with the Fibonacci sequence.
As the Fibonacci sequence progresses, the ratio of consecutive terms increasingly approximates the golden ratio.
The golden ratio, (1 + sqrt(5))/2, is approximately 1.61803398875. It is an irrational number.

Binet’s formula

The nth Fibonacci number can be expressed in a closed form using Binet’s formula: f(n) = (Φ^n - (-Φ)^-n) / sqrt(5).
Remember that in Binet’s formula, Φ is the golden ratio and sqrt(5) is the square root of 5.

Pell Numbers

In addition to the Fibonacci sequence, there are many other similar sequences with different initial conditions or recurrence relations.
One of such sequences is the Pell numbers or Pell’s sequence. It is defined as p(0) = 0, p(1) = 1, and p(n) = 2*p(n-1) + p(n-2) for n > 1.

Lucas Numbers

Another similar sequence is the Lucas numbers. This sequence is defined as l(0) = 2, l(1) = 1, and l(n) = l(n-1) + l(n-2) for n > 1.
The Lucas numbers have similar properties to the Fibonacci sequence but the initial values are different.

The Fibonacci sequence and other sequences with similar recurrence relations can be seen in numerous areas including computer algorithms, counting problems, and biological modelling.
The Fibonacci sequence also has interesting relationships with continued fractions, fractals, and Euclidean algorithm.
The golden ratio, associated with the Fibonacci sequence, has many surprising appearances in art, architecture, and nature.