The Fibonacci sequence is a sequence of numbers where each term is the sum of the two terms before it: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144... Starting from 0 and 1, every subsequent number is generated by adding the previous two.
Building the sequence. 0+1=1. 1+1=2. 1+2=3. 2+3=5. 3+5=8. 5+8=13. 8+13=21. And so on, indefinitely — each new term only requires knowing the two immediately before it, not the whole history of the sequence.
Named After, but Not Discovered by, Fibonacci
The sequence is named for Leonardo of Pisa, known as Fibonacci, who introduced it to Western European mathematics in 1202 through a problem about rabbit population growth in his book Liber Abaci. However, the sequence had already been described centuries earlier in Indian mathematics, particularly in the context of Sanskrit poetic meter (counting the number of ways to construct a rhythm of a given length from short and long syllables) — an example of the same mathematical pattern being independently discovered for genuinely different practical reasons in different cultures.
Checking whether a number belongs to the Fibonacci sequence. A number n is a Fibonacci number if and only if at least one of (5n²+4) or (5n²−4) is a perfect square — a somewhat surprising but provable mathematical fact that avoids needing to generate the whole sequence up to n just to check.
Worked example: is 21 a Fibonacci number? Check 5×21²+4 = 5×441+4 = 2209. Is 2209 a perfect square? √2209 = 47 exactly. Yes, so 21 is a Fibonacci number (which checks out — it's the 8th term in the sequence above).
Worked example: is 22 a Fibonacci number? Check 5×22²+4 = 2424+4=2424... let's compute carefully: 22²=484, 5×484=2420, +4=2424. √2424 ≈ 49.23, not a whole number. Check the other formula: 5×484−4=2416. √2416 ≈ 49.15, also not whole. Neither check produces a perfect square, so 22 is not a Fibonacci number.
The Golden Ratio Connection
As you go further into the Fibonacci sequence, the ratio between consecutive terms gets closer and closer to a specific irrational number called the golden ratio, approximately 1.618. 21÷13 ≈ 1.615. 34÷21 ≈ 1.619. 55÷34 ≈ 1.618. The ratio never lands on the golden ratio exactly at any finite term, but converges toward it as the sequence continues indefinitely — a genuine, provable mathematical relationship, not a numerological coincidence.
Where Fibonacci Numbers Genuinely Show Up in Nature
Fibonacci numbers do show up in real biological structures, especially in the spiral arrangements of seeds in a sunflower head, the scales of a pinecone, and the branching patterns of some plants — this is a well-documented phenomenon in botany, related to how plants efficiently pack new growth around a central point (phyllotaxis). However, popular claims that Fibonacci numbers or the golden ratio govern the proportions of the human body, famous artworks, or architecture with mathematical precision are, in most specific cases, considerably overstated or based on cherry-picked measurements — worth knowing so as not to repeat an overreaching claim as settled fact.
A related sequence worth knowing: Lucas numbers. The Lucas sequence follows the identical addition rule (each term is the sum of the two before it) but starts from 2 and 1 instead of 0 and 1: 2, 1, 3, 4, 7, 11, 18, 29... It shares many of the Fibonacci sequence's mathematical properties, including converging to the same golden ratio, which illustrates that the "add the previous two" rule itself — not the specific starting values 0 and 1 — is what produces most of the sequence's famous properties.
Common mistakes. Forgetting where the sequence starts is a common source of off-by-one errors — some sources start counting from 0, 1, 1, 2... and others start from 1, 1, 2, 3..., which shifts every term's position number by one depending on convention. A second common mistake is assuming any coincidental appearance of a Fibonacci-adjacent number in nature or art proves an underlying "Fibonacci law" at work, when the connection may just be coincidental for that specific case.
Worked example: verify that consecutive Fibonacci numbers are always coprime (share no common factor besides 1), using 21 and 34. GCF(21,34): 21=3×7, 34=2×17 — no shared prime factors, so GCF=1, confirming they're coprime. This holds for every consecutive pair in the sequence, a provable general property rather than a coincidence specific to this one pair, and it's part of why Fibonacci numbers show up in certain number-theoretic proofs about coprimality.
Worked example: use the golden ratio directly to estimate a Fibonacci term without building the whole sequence. For large n, F(n) is closely approximated by φⁿ/√5, where φ≈1.618 is the golden ratio. Estimating F(15): φ¹⁵/√5 ≈ 1364.0/2.236 ≈ 610 — matching the actual 15th term exactly once rounded, since this approximation (Binet's formula) becomes extremely accurate for larger n.
Worked example: build the sequence out ten more terms past 144, to see the growth rate. Continuing from 89, 144: 89+144=233, 144+233=377, 233+377=610, 377+610=987, 610+987=1597, 987+1597=2584, 1597+2584=4181, 2584+4181=6765, 4181+6765=10946, 6765+10946=17711. Notice how quickly the terms grow relative to how few steps it takes — Fibonacci growth is exponential in the long run (tracking the golden ratio raised to increasing powers), even though each individual step is just a simple addition.
Worked example: is 100 a Fibonacci number? Check 5×100²+4 = 50,004. √50,004 ≈ 223.6, not a whole number. Check 5×100²−4 = 49,996. √49,996 ≈ 223.6, also not whole. Neither is a perfect square, so 100 is not a Fibonacci number — consistent with the actual sequence, which jumps from 89 to 144, skipping over 100 entirely.
Fibonacci numbers and computer science. Beyond biology, Fibonacci numbers show up in computer science in the analysis of certain algorithms (notably the time complexity of naive recursive Fibonacci computation itself, a classic teaching example of exponential-versus-linear algorithm design) and in Fibonacci search techniques, an alternative to binary search in specific contexts.
Negafibonacci, a lesser-known extension worth a brief mention. The addition rule can be run backward as well as forward, extending the sequence to negative indices: F(−1)=1, F(−2)=−1, F(−3)=2, F(−4)=−3, alternating in sign. This isn't something most people ever need practically, but it's a good illustration of how a simple recursive rule can be extended cleanly in directions beyond its original, intuitive definition.
For a differently-patterned but similarly famous sequence, see What Are Triangular Numbers?