The Fibonacci Sequence is a series of numbers where each number $F(n)$ is the sum of the two preceding ones, often starting with 0 and 1. That is:

$$F(n) = F(n-1) + F(n-2)$$

with initial conditions

$$F(0) = 0, \quad F(1) = 1$$

Golden Ratio

The ratio of consecutive Fibonacci numbers approximates the Golden Ratio ($\phi \approx 1.6180339887$):

$$\lim_{{n \to \infty}} \frac{{F(n+1)}}{{F(n)}} = \phi$$

Real-World Occurrences

The sequence frequently manifests in nature, such as in flower petals, seedhead spirals, and seashell growth patterns.

Visual Representation

Code Example: Calculating The Nth Fibonacci Number

Here is the Python code:

# Using recursion
def fibonacci_recursive(n):
    if n <= 0: return 0
    elif n == 1: return 1
    return fibonacci_recursive(n-1) + fibonacci_recursive(n-2)
    
# Using dynamic programming for efficiency
def fibonacci_dynamic(n):
    fib = [0, 1]
    for i in range(2, n+1):
        fib.append(fib[-1] + fib[-2])
    return fib[n]