11.7.2. Fibonacci Series

The Fibonacci sequence is a series of numbers where the next number is the sum of the previous two: • 𝐹₀ = 0 • 𝐹₁ = 1 • 𝐹ₙ = 𝐹ₙ₋₁ + 𝐹ₙ₋₂ for 𝑛 ≥ 2

The Fibonacci sequence starts with two initial numbers, 0 and 1. Each subsequent number in the series is generated by adding the two numbers before it. For instance, the numbers after 0 and 1 are calculated as follows: 0 + 1 = 1 (the next number), then 1 + 1 = 2, followed by 1 + 2 = 3, and so on. This continues indefinitely, creating a sequence of numbers that have fascinating properties and appear in various natural phenomena.

Think of the Fibonacci sequence like a pattern of branching in trees. If each branch splits into two smaller branches, the growth of those branches can be described by Fibonacci numbers. In nature, this sequence can be observed in the arrangement of leaves, flower petals, or even the pattern of seeds in a sunflower.

Python code example:

def fibonacci(n):
    if n == 0:
        return 0 # base case
    elif n == 1:
        return 1 # base case
    else:
        return fibonacci(n-1) + fibonacci(n-2) # recursive case
# Example usage:
print(fibonacci(7)) # Output: 13

The provided Python code demonstrates how to implement the Fibonacci sequence using recursion. The function fibonacci(n) takes an integer n as input and returns the nth Fibonacci number. The function first checks the base cases: if n is 0, it returns 0, and if n is 1, it returns 1. For any other value of n, the function makes two recursive calls to itself, adding the results of fibonacci(n-1) and fibonacci(n-2). This approach effectively breaks the problem down into smaller components until it reaches the base cases. As indicated in the example usage, calling fibonacci(7) will ultimately return 13.

Imagine you are determining how many pairs of rabbits you can breed in a year, starting with a pair that can reproduce every month. Each month, each pair produces another pair, but those new pairs do not themselves reproduce until they reach maturity in the second month. If you model the growth of the rabbit pairs each month using the Fibonacci sequence, you will find that by the seventh month, you have 13 pairs. This is the same as tracing the Fibonacci sequence to n=7.

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