Fibonacci Sequence

IT 위키

Fibonacci Sequence is a mathematical sequence where each number is the sum of the two preceding numbers. It is widely used in mathematics, computer science, and nature to model growth patterns and recursive structures.

1 Definition[편집 | 원본 편집]

The Fibonacci sequence is defined recursively as:

  • F(0) = 0, F(1) = 1 (Base cases)
  • F(n) = F(n-1) + F(n-2) for n ≥ 2

2 Example Sequence[편집 | 원본 편집]

The first few Fibonacci numbers are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

3 Closed-Form Formula (Binet's Formula)[편집 | 원본 편집]

The nth Fibonacci number can be computed directly using Binet’s formula:

  • F(n) = (φⁿ - ψⁿ) / √5

where:

  • φ = (1 + √5) / 2 (Golden Ratio)
  • ψ = (1 - √5) / 2

4 Properties[편집 | 원본 편집]

  • Recurrence Relation: Each term is the sum of the previous two.
  • Golden Ratio Connection: The ratio of consecutive Fibonacci numbers converges to φ.
  • Parity Pattern: Even numbers appear at every third position.
  • Sum of Fibonacci Numbers: The sum of the first n Fibonacci numbers is F(n+2) - 1.

5 Computational Methods[편집 | 원본 편집]

There are several ways to compute Fibonacci numbers:

  • Recursive Approach:
    • Uses the recurrence relation directly.
    • Complexity: O(2ⁿ) (Exponential, inefficient for large n).
  • Iterative Approach:
    • Uses a loop to compute Fibonacci numbers.
    • Complexity: O(n) (Linear, more efficient than recursion).
  • Matrix Exponentiation:
    • Uses matrix multiplication to compute Fibonacci numbers efficiently.
    • Complexity: O(log n).
  • Binet’s Formula:
    • Uses the closed-form expression.
    • Complexity: O(1), but limited by floating-point precision.

6 Applications[편집 | 원본 편집]

Fibonacci numbers appear in various fields:

  • Computer Science: Recursive algorithm analysis, dynamic programming.
  • Mathematics: Number theory, combinatorial counting.
  • Nature: Leaf arrangements, branching patterns in plants.
  • Financial Markets: Fibonacci retracement in technical analysis.
  • Art and Architecture: Proportions related to the golden ratio.

7 Comparison of Computational Methods[편집 | 원본 편집]

Method Approach Complexity
Recursion Direct recursive relation O(2ⁿ)
Iteration Loop-based calculation O(n)
Matrix Exponentiation Fast exponentiation of Fibonacci matrix O(log n)
Binet’s Formula Direct computation using golden ratio O(1)

8 See Also[편집 | 원본 편집]