Fibonacci Calculator

1. Generate a sequence. Enter how many Fibonacci numbers you want, up to 50. The calculator shows F(0) through F(n-1), starting at 0. It also shows the Nth term and the golden ratio approximation formed by dividing consecutive terms.
2. Check any number. Switch to the "Check a Number" tab and enter any non-negative integer. The calculator instantly tells you whether it is a Fibonacci number using the closed-form test (no need to generate the sequence).
3. Golden ratio. As n grows, the ratio F(n)/F(n-1) converges to φ = 1.6180339887... The calculator shows both the approximation from your sequence and the true value for comparison. By F(20), the ratio is accurate to 7 decimal places.

Example: the first 10 Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34. The ratio 34/21 = 1.619047..., which is already very close to φ.

Recursive definition:
  F(0) = 0
  F(1) = 1
  F(n) = F(n-1) + F(n-2)  for n ≥ 2

First 15 terms:
  n:    0  1  1  2  3  5  8  13  21  34  55  89  144  233  377

Closed-form (Binet's formula):
  F(n) = (φⁿ - ψⁿ) / √5
  where φ = (1+√5)/2 ≈ 1.6180339887 (golden ratio)
        ψ = (1-√5)/2 ≈ -0.6180339887

Golden ratio relationship:
  lim(n→∞) F(n+1)/F(n) = φ = 1.6180339887...

Test if N is a Fibonacci number:
  N is Fibonacci iff (5N²+4) or (5N²-4) is a perfect square.
  Example: N=34: 5×34²+4 = 5780+4 = 5784. √5784 ≈ 76.05 (no)
           5×34²-4 = 5776. √5776 = 76. 76² = 5776. Yes! 34 is Fibonacci.

Sum identity: F(0)+F(1)+...+F(n) = F(n+2) - 1
  Example: 0+1+1+2+3+5 = 12 = F(7)-1 = 13-1