Fibonacci Sequence 편집하기

'''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.
==Definition==
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
==Example Sequence==
The first few Fibonacci numbers are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
==Closed-Form Formula (Binet's Formula)==
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
==Properties==
*'''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.
==Computational Methods==
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.
==Applications==
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.
==Comparison of Computational Methods==
{| class="wikitable"
!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)
|}
==See Also==
*[[Recurrence Relation]]
*[[Golden Ratio]]
*[[Dynamic Programming]]
*[[Algorithm Complexity]]
*[[Lucas Numbers]]
*[[Pascal’s Triangle]]
[[Category:Algorithm]]