Fibonacci Calculator

The Fibonacci sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... where each number is the sum of the two preceding numbers.

The golden ratio φ = (1 + √5)/2 ≈ 1.618. The ratio of consecutive Fibonacci numbers approaches φ as n increases.

Use the recursive formula F(n) = F(n-1) + F(n-2), or Binet's formula for direct computation.

F(n) = (φⁿ - ψⁿ)/√5, where φ = (1+√5)/2 and ψ = (1-√5)/2. This gives the exact value without recursion.

They appear in flower petal counts, pinecone spirals, shell spirals, leaf arrangements, and tree branching patterns.

No, the sequence is infinite. However, Fibonacci numbers grow exponentially (approximately 1.618ⁿ).

The ratio F(n+1)/F(n) converges to φ as n increases. This connection is fundamental to Fibonacci's properties.

Fibonacci retracement levels (23.6%, 38.2%, 50%, 61.8%) are used in technical analysis to predict support and resistance levels.

A logarithmic spiral approximated by drawing quarter-circle arcs through the squares of Fibonacci numbers. It appears in nautilus shells and galaxies.

Yes, the sequence can be extended to negative indices: F(-n) = (-1)ⁿ⁺¹ × F(n). This is called the Negafibonacci sequence.

Naive recursion is O(2ⁿ). Dynamic programming or matrix exponentiation reduces it to O(n) or O(log n).

Zeckendorf's theorem states every positive integer can be uniquely represented as a sum of non-consecutive Fibonacci numbers.

The golden spiral is approximated by connecting quarter-circles in Fibonacci-sized squares. It appears in nature and art.

Lucas numbers follow the same recurrence as Fibonacci but start with 2, 1 instead of 0, 1. They are closely related to Fibonacci numbers.

Fibonacci numbers that are also prime. Known Fibonacci primes include F(3)=2, F(4)=3, F(5)=5, F(7)=13, F(11)=89, F(13)=233.

Fibonacci heaps are used in graph algorithms. Fibonacci search is a search algorithm. The sequence itself demonstrates dynamic programming.

Σ F(i) from i=1 to n = F(n+2) - 1. For example, 1+1+2+3+5 = 12 = F(7) - 1.

Yes, using modular arithmetic, matrix exponentiation, or fast doubling algorithms. Numbers grow exponentially, so large values need big integer support.

The Pisano period is the period of Fibonacci numbers modulo m. For example, Fibonacci mod 3 repeats every 8 terms.

Applications include Fibonacci heaps, Fibonacci search, dynamic programming examples, and analysis of recursive algorithms.

The Fibonacci Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Fibonacci Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.