55
34
89
123
1.61764706
55
1.421085471520e-14
0
0
55
34
89
123
1.61764706
55
1.421085471520e-14
0
0
The Fibonacci Numbers Calculator computes the $$n$$-th Fibonacci number $$F_n$$ using Binet's formula, the elegant closed-form expression involving the golden ratio. The Fibonacci sequence is defined by the recurrence $$F_0 = 0$$, $$F_1 = 1$$, and $$F_n = F_{n-1} + F_{n-2}$$ for $$n \geq 2$$, producing the sequence $$0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \ldots$$
Leonardo of Pisa, known as Fibonacci, introduced this sequence to Western mathematics in his 1202 work Liber Abaci, using it to model rabbit population growth. However, the sequence was known centuries earlier to Indian mathematicians — Virahanka (c. 700 CE) and Hemachandra (c. 1150 CE) encountered it in the study of Sanskrit prosody and the enumeration of poetic meters.
Binet's formula provides a direct computation without recursion:
$$F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}$$
where $$\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.6180$$ is the golden ratio and $$\psi = \frac{1 - \sqrt{5}}{2} \approx -0.6180$$ is its conjugate. Since $$|\psi| < 1$$, the term $$\psi^n$$ vanishes rapidly, so $$F_n \approx \varphi^n / \sqrt{5}$$ for large $$n$$. Jacques Philippe Marie Binet published this formula in 1843, though Abraham de Moivre and Euler had known equivalent results earlier.
The ratio of consecutive Fibonacci numbers $$F_n / F_{n-1}$$ converges to the golden ratio $$\varphi = 1.618033988\ldots$$, and this convergence is the slowest possible among all continued fraction convergents — making $$\varphi$$ the "most irrational" number. This property explains why sunflower seed spirals, pinecone patterns, and phyllotaxis in plants follow Fibonacci-related angles: the golden angle $$360°/\varphi^2 \approx 137.5°$$ produces the most uniform packing.
The calculator also computes the corresponding Lucas number $$L_n = \varphi^n + \psi^n$$, the companion sequence $$2, 1, 3, 4, 7, 11, 18, \ldots$$ that shares the same recurrence but different initial conditions. Fibonacci and Lucas numbers are intimately linked: $$L_n = F_{n-1} + F_{n+1}$$ and $$F_{2n} = F_n L_n$$.
The calculator uses Binet's closed-form formula to compute Fibonacci numbers directly without iteration.
Step 1: Define the golden ratio and its conjugate.
$$\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887 \quad \psi = \frac{1 - \sqrt{5}}{2} \approx -0.6180339887$$
These are the roots of the characteristic equation $$x^2 = x + 1$$ associated with the Fibonacci recurrence.
Step 2: Apply Binet's formula.
$$F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}$$
The result is always an integer despite the irrational numbers involved.
Step 3: Compute the Lucas number.
$$L_n = \varphi^n + \psi^n$$
Step 4: Compute the consecutive ratio. $$F_n / F_{n-1}$$ converges to $$\varphi$$.
Parity pattern: Every third Fibonacci number is even ($$F_0, F_3, F_6, \ldots$$), because the pattern even-odd-odd repeats with period 3. This follows from the recurrence: even+odd=odd, odd+odd=even.
Fₙ is the $$n$$-th Fibonacci number. $$F_{10} = 55$$, $$F_{20} = 6765$$, $$F_{30} = 832040$$.
Fₙ₋₁ and Fₙ₊₁ are the neighboring values. Their sum equals $$L_n$$, the corresponding Lucas number.
Fₙ/Fₙ₋₁ approximates the golden ratio $$\varphi$$. The convergence is slow — the approximation alternates above and below $$\varphi$$ — which is related to the continued fraction expansion $$\varphi = [1; 1, 1, 1, \ldots]$$.
Binet Formula Value shows the unrounded real number from Binet's formula. The fractional part should be very close to 0 or 1 (rounding to the exact integer).
Lucas Number Lₙ satisfies the same recurrence as Fibonacci but starts $$L_0 = 2, L_1 = 1$$. Many Fibonacci identities have Lucas analogues.
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The ratio 55/34 ≈ 1.6176 already approximates φ to 3 decimal places. The Lucas number L₁₀ = 123 = F₉ + F₁₁ = 34 + 89.
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By n=20, the ratio F₂₀/F₁₉ = 6765/4181 ≈ 1.61803399 matches φ to 8 significant figures. The golden ratio emerges naturally from the Fibonacci recurrence.
If the ratio converges to some limit $$L$$, then from the recurrence $$F_n = F_{n-1} + F_{n-2}$$, dividing by $$F_{n-1}$$: $$L = 1 + 1/L$$, giving $$L^2 = L + 1$$, whose positive root is $$\varphi = (1+\sqrt{5})/2$$. Binet's formula makes this precise: $$F_n/F_{n-1} = (\varphi^n - \psi^n)/(\varphi^{n-1} - \psi^{n-1}) \to \varphi$$ since $$|\psi/\varphi| < 1$$.
JavaScript's 64-bit floating-point numbers represent integers exactly up to $$2^{53} \approx 9 \times 10^{15}$$. $$F_{70} = 190,392,490,709,135$$ is within this range. Beyond $$n \approx 75$$, rounding errors in Binet's formula may introduce inaccuracies. For exact computation of very large Fibonacci numbers, arbitrary-precision integer arithmetic is needed.
Fibonacci numbers appear in the spiral arrangements of sunflower seeds, pinecone scales, and pineapple fruitlets (phyllotaxis). The number of petals on many flowers follows Fibonacci numbers (3 for lilies, 5 for buttercups, 8 for delphiniums, 13 for marigolds). This occurs because the golden angle $$\approx 137.5°$$ produces optimal packing, and this angle is intimately connected to the golden ratio.
Key identities include: $$L_n = F_{n-1} + F_{n+1}$$, $$F_{2n} = F_n L_n$$, $$L_n^2 - 5F_n^2 = 4(-1)^n$$ (a Pell-like equation), and $$F_n = (L_{n-1} + L_{n+1})/5$$. Both sequences satisfy the same recurrence $$x_n = x_{n-1} + x_{n-2}$$ but with different seeds.
Zeckendorf's theorem states that every positive integer can be uniquely represented as a sum of non-consecutive Fibonacci numbers. For example, $$20 = 13 + 5 + 2 = F_7 + F_5 + F_3$$. This greedy algorithm representation is analogous to binary representation but uses Fibonacci numbers as the base. It's used in Fibonacci coding for data compression.
$$\gcd(F_m, F_n) = F_{\gcd(m,n)}$$ — the GCD of two Fibonacci numbers is itself a Fibonacci number, indexed by the GCD of the original indices. This elegant identity means that $$F_m | F_n$$ if and only if $$m | n$$. Also, computing $$\gcd(F_n, F_{n-1})$$ via the Euclidean algorithm takes exactly $$n$$ steps, making consecutive Fibonacci numbers the worst case for the algorithm.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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