F(10)
55
Terms
11
φ (phi)
1.618034
Enter 0–78. Generates the sequence F(0) through F(n).
Uses the property that n is Fibonacci if 5n²±4 is a perfect square.
55
1.61803399
55
F(n) = F(n-1) + F(n-2), with F(0) = 0, F(1) = 1Each Fibonacci number is the sum of the two preceding numbers in the sequence.
Where:
F(n)= The nth Fibonacci numbern= Position in the sequence (0-indexed)F(n) = (φⁿ - ψⁿ) / √5Direct formula using the golden ratio, avoiding iterative computation.
Where:
φ= (1 + √5) / 2 ≈ 1.618 (golden ratio)ψ= (1 - √5) / 2 ≈ -0.618 (conjugate)√5= Square root of 5 ≈ 2.236φ = (1 + √5) / 2 ≈ 1.6180339887...The limit of F(n+1)/F(n) as n approaches infinity. An irrational number with unique mathematical properties.
Where:
φ= Phi, the golden ratio√5= Square root of 5Inputs
Result
F(10) = F(9) + F(8) = 34 + 21 = 55. The ratio 55/34 ≈ 1.6176 is close to the golden ratio 1.6180.
Inputs
Result
F(20) = 6,765. The ratio F(20)/F(19) = 6765/4181 = 1.6180339... matches the golden ratio to 7 decimal places.
Inputs
Result
5 × 144² + 4 = 5 × 20,736 + 4 = 103,684 = 322². Since this is a perfect square, 144 is a Fibonacci number (F(12)).
The Fibonacci sequence starts with 0 and 1, and each subsequent number is the sum of the two preceding ones: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on. It was introduced to Western mathematics by Leonardo of Pisa (Fibonacci) in 1202.
| n | F(n) | Calculation |
|---|---|---|
| 0 | 0 | Base case |
| 1 | 1 | Base case |
| 5 | 5 | 3 + 2 |
| 10 | 55 | 34 + 21 |
| 20 | 6,765 | 4,181 + 2,584 |
The golden ratio (phi, φ) equals (1 + √5) / 2, approximately 1.6180339887. The ratio of consecutive Fibonacci numbers F(n+1)/F(n) converges to phi as n increases. By F(10)/F(9) = 55/34 = 1.6176..., it is already accurate to 3 decimal places.
| Ratio | Value | Error from φ |
|---|---|---|
| F(5)/F(4) = 5/3 | 1.6667 | 0.0486 |
| F(8)/F(7) = 21/13 | 1.6154 | 0.0027 |
| F(10)/F(9) = 55/34 | 1.6176 | 0.0004 |
| F(15)/F(14) = 610/377 | 1.61803 | 0.00001 |
Binet's formula computes F(n) directly without iteration: F(n) = (φ^n - ψ^n) / √5, where φ = (1+√5)/2 and ψ = (1-√5)/2. Since |ψ| < 1, the ψ^n term shrinks rapidly, so F(n) ≈ round(φ^n / √5) for n >= 0.
| n | Iterative F(n) | Binet Formula | Match |
|---|---|---|---|
| 5 | 5 | 5.000 | Yes |
| 10 | 55 | 55.000 | Yes |
| 20 | 6,765 | 6,765.000 | Yes |
| 30 | 832,040 | 832,040.000 | Yes |
A positive integer n is a Fibonacci number if and only if 5n² + 4 or 5n² - 4 is a perfect square. For example, 13: 5(169) + 4 = 849 (not square), 5(169) - 4 = 841 = 29² (perfect square), so 13 is Fibonacci.
| Number | 5n²+4 | 5n²-4 | Fibonacci? |
|---|---|---|---|
| 8 | 324 = 18² | 316 | Yes |
| 13 | 849 | 841 = 29² | Yes |
| 14 | 984 | 976 | No |
| 21 | 2209 = 47² | 2201 | Yes |
Fibonacci numbers appear in many natural phenomena: the number of petals on flowers (lilies have 3, buttercups 5, daisies 13 or 21), spiral patterns in sunflower heads and pinecones, and the branching of trees. The golden spiral based on Fibonacci rectangles models nautilus shells.
| Natural Phenomenon | Fibonacci Numbers | Pattern |
|---|---|---|
| Lily petals | 3 | Petal count |
| Sunflower | 34, 55 | Opposing spirals |
| Pinecone | 8, 13 | Opposing spirals |
| Pineapple | 8, 13, 21 | Hexagonal scales |
Last Updated: Mar 9, 2026
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