Understanding the Fibonacci Series

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones. It typically starts with:

0, 1, 1, 2, 3, 5, 8, 13, ...

The Basic Recursive Approach

The classic recursive method for calculating Fibonacci numbers is:

FUNCTION Fibonacci(n):
    IF n <= 1 THEN
        RETURN n
    END IF
    RETURN Fibonacci(n-1) + Fibonacci(n-2)

This approach results in two recursive calls for each number, making it inefficient for larger numbers.

Introducing the LinearFibonacci Algorithm

The LinearFibonacci method offers an optimization by returning two consecutive Fibonacci numbers during each recursive call.

Algorithm:

Input: A non-negative integer k.
Output: A pair of consecutive Fibonacci numbers (Fib(k), Fib(k-1)).

FUNCTION LinearFibonacci(k):
    IF k <= 1 THEN
        RETURN (k, 0)
    END IF
    (i, j) = LinearFibonacci(k-1)
    RETURN (i+j, i)

This method reduces the overall recursive calls by returning two values at once.