fibonacci

Calculates the nth Fibonacci number with optimized iterative algorithm and caching. Computes the Fibonacci sequence F(n) where F(0) = 0, F(1) = 1, and F(n) = F(n-1) + F(n-2) for n > 1. Uses an efficient iterative approach with O(n) time complexity and O(1) space complexity for the calculation, plus intelligent caching for performance optimization.

Signature

const fibonacci: (number: number) => number

Parameters

Name	Type	Description
number	-	Position in Fibonacci sequence (must be non-negative integer)

Returns

The nth Fibonacci number

Examples

Basic Fibonacci calculations

import { fibonacci } from '@winglet/common-utils';

console.log(fibonacci(0)); // 0 (first Fibonacci number)
console.log(fibonacci(1)); // 1 (second Fibonacci number)
console.log(fibonacci(2)); // 1 (0 + 1)
console.log(fibonacci(3)); // 2 (1 + 1)
console.log(fibonacci(8)); // 21 (sequence: 0,1,1,2,3,5,8,13,21)
console.log(fibonacci(15)); // 610

Performance optimization and large numbers

// First calculation computes and caches intermediate results
console.log(fibonacci(50)); // 12586269025 (computed and cached)

// Subsequent calls for smaller numbers use cached values
console.log(fibonacci(45)); // Retrieved from cache or computed efficiently

// Large Fibonacci numbers (JavaScript integer limit considerations)
console.log(fibonacci(78)); // 8944394323791464 (near safe integer limit)

Playground

import { fibonacci } from '@winglet/common-utils';

console.log(fibonacci(0)); // 0 (first Fibonacci number)
console.log(fibonacci(1)); // 1 (second Fibonacci number)
console.log(fibonacci(2)); // 1 (0 + 1)
console.log(fibonacci(3)); // 2 (1 + 1)
console.log(fibonacci(8)); // 21 (sequence: 0,1,1,2,3,5,8,13,21)
console.log(fibonacci(15)); // 610

Notes

Mathematical Properties:

• F(0) = 0, F(1) = 1 by definition
• Each number is the sum of the two preceding ones
• Golden ratio φ ≈ 1.618 is related to Fibonacci ratios: F(n+1)/F(n) → φ
• Growth rate is exponential: F(n) ≈ φⁿ/√5 (Binet's formula)

Use Cases:

• Mathematical modeling and algorithm analysis
• Nature simulations (spiral patterns, growth models)
• Financial market analysis (Fibonacci retracements)
• Computer graphics and generative art
• Algorithm optimization and dynamic programming examples
• Teaching recursion and iterative optimization

Performance: O(n) time complexity for uncached calculations, O(1) for cached results. Space complexity is O(1) for the algorithm plus cache storage. More efficient than naive recursive approaches which have O(φⁿ) complexity.