Question 1

What is the Fibonacci sequence?

Accepted Answer

It is the sequence of integers in which each term is the sum of the two before it: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34... It is defined by F(n) = F(n−1) + F(n−2), with F(0) = 0 and F(1) = 1. Leonardo of Pisa, known as Fibonacci, introduced it in Europe in 1202 in the Liber Abaci, in a problem about rabbit breeding. The sequence was already known in Indian mathematics, in studies of poetic meter.

Question 2

Does the sequence start at F(0) = 0 or F(1) = 1?

Accepted Answer

It depends on the convention, and both describe the same values. The modern definition, and the one programs use, has F(0) = 0 and F(1) = 1, which makes identities such as F(m+n) cleaner and matches array indices that start at 0. Older texts start at 1, 1, 2, 3, 5, shifting the index. In the Sequence mode, the Start at switch lets you choose; just check which convention your book or exercise uses.

Question 3

What is Binet's formula and why does it fail for large n?

Accepted Answer

Binet's formula is F(n) = (φⁿ − ψⁿ) / √5, with φ = (1 + √5)/2 and ψ = (1 − √5)/2. It is exact in mathematics. In 64-bit floating point (double), however, φⁿ gains more digits than the mantissa keeps, and rounding gives the wrong integer from F(71) on. Near n = 1475, φⁿ overflows to infinity. With BigInt, by addition or doubling, the result is exact. The Nth term mode shows both versions side by side.

Question 4

How are Fibonacci numbers related to the golden ratio?

Accepted Answer

The ratio F(n+1)/F(n) converges to the golden ratio φ = (1 + √5)/2 ≈ 1.6180339887. The error shrinks by a factor of about 0.382 (1/φ²) per step, and the ratio sits alternately above and below φ. 55/34 ≈ 1.6176 is already close, and by n = 30 about 12 decimal places agree. Convergence holds for any pair of non-zero integer seeds with the sum rule, not just 0 and 1.

Question 5

Does Fibonacci really show up in nature?

Accepted Answer

In some cases yes, but with caveats. In sunflowers, pine cones and pineapples, the number of spirals in each direction is often two consecutive Fibonacci numbers, and the angle between successive leaves in many plants is near 137.5°, the golden angle. Packing models explain why that angle is efficient. But there are exceptions, and it is not true that everything in nature follows Fibonacci: the nautilus shell, for example, is a logarithmic spiral whose growth ratio is not the golden one.

Question 6

Does Fibonacci have applications in computing?

Accepted Answer

Yes, in several places. The Fibonacci heap, by Fredman and Tarjan, is a priority queue with constant amortized insert and decrease-key; the name comes from the analysis, where tree sizes are bounded by Fibonacci numbers. Fibonacci search splits a sorted array into Fibonacci-sized parts and is also logarithmic. In AVL trees, the minimum number of nodes for a given height grows like Fibonacci. And naive recursive fib(n) is the classic example of exponential complexity: fib(30) makes 2,692,537 calls.

Question 7

How can I compute F(n) fast, without adding n times?

Accepted Answer

With fast doubling, which uses F(2k) = F(k)·(2·F(k+1) − F(k)) and F(2k+1) = F(k)² + F(k+1)². Each step doubles the index, so F(n) takes about log₂(n) big-integer multiplication steps instead of n additions. It is equivalent to exponentiating the matrix [[1,1],[1,0]]. This tool computes F(100000), with 20899 digits, with that method and shows the time measured in your browser.

Question 8

How do I tell whether a number is a Fibonacci number?

Accepted Answer

An integer n ≥ 0 is a Fibonacci number if and only if 5n² + 4 or 5n² − 4 is a perfect square. The tool applies that test with a BigInt integer square root, so it works even for 1000-digit numbers. If the number is Fibonacci, it shows the index. If not, it shows the neighbouring terms and the Zeckendorf decomposition: every positive integer is, uniquely, a sum of non-consecutive Fibonacci numbers, such as 100 = 89 + 8 + 3.

Question 9

What is the Pisano period and what is F(n) mod m for?

Accepted Answer

The residues of F(n) mod m form a periodic sequence, and the length of the cycle is the Pisano period π(m). For example, π(2) = 3, π(3) = 8, π(5) = 20 and π(10) = 60, meaning the last digit of Fibonacci numbers repeats every 60 terms. π(m) never exceeds 6m. This lets you find F(n) mod m for a gigantic n by reducing n modulo π(m), common in competitive programming problems. The tool does both computations and checks that they agree.

Question 10

How far does the tool go, and does my data leave the browser?

Accepted Answer

The term list goes up to 2000, the Nth term goes up to n = 100000, the Is it Fibonacci? test accepts up to 1000 digits, and the Pisano period goes up to m = 10000. Everything uses BigInt, with no floating-point rounding in the values. The calculations run in your browser and no number you type is sent to a server.

Index	Value	Parity
F(0)	0	even
F(1)	1	odd
F(2)	1	odd
F(3)	2	even
F(4)	3	odd
F(5)	5	odd
F(6)	8	even
F(7)	13	odd
F(8)	21	odd
F(9)	34	even
F(10)	55	odd
F(11)	89	odd
F(12)	144	even
F(13)	233	odd
F(14)	377	odd
F(15)	610	even
F(16)	987	odd
F(17)	1597	odd
F(18)	2584	even
F(19)	4181	odd

Index	Value	Parity
F(0)	0	even
F(1)	1	odd
F(2)	1	odd
F(3)	2	even
F(4)	3	odd
F(5)	5	odd
F(6)	8	even
F(7)	13	odd
F(8)	21	odd
F(9)	34	even
F(10)	55	odd
F(11)	89	odd
F(12)	144	even
F(13)	233	odd
F(14)	377	odd
F(15)	610	even
F(16)	987	odd
F(17)	1597	odd
F(18)	2584	even
F(19)	4181	odd

Fibonacci Sequence Generator