User Rating 0.0 ★★★★★

Total Usage 0 times

Category Math & Algebra

Order (n)

Range: 0–40. Words grow exponentially.

Seed S(0) Default: 0

Seed S(1) Default: 01

Fibonacci Words

n	Length	Density "0"	Word

Is this tool helpful?

Your feedback helps us improve.

★ ★ ★ ★ ★

About

A Fibonacci word is an infinite binary string constructed by repeated concatenation: S(n) = S(n − 1) ⋅ S(n − 2), seeded with S(0) = 0 and S(1) = 01. The length of each word follows the Fibonacci number sequence exactly. The density of the character "0" converges to 1/φ ≈ 0.6180, where φ is the golden ratio. Fibonacci words appear in quasicrystal modeling, aperiodic tilings, and lossless data compression research. Miscounting the concatenation order or swapping operands produces a fundamentally different string family with incorrect fractal dimension properties.

This tool computes Fibonacci words up to order 40, reports exact lengths, character frequencies, and density convergence. Custom seed words are supported for generalized Fibonacci string systems. Note: word lengths grow exponentially. At order 40, the string contains over 165 million characters. The tool computes full metadata but truncates display beyond 10,000 characters to preserve browser responsiveness.

Formulas

The classical Fibonacci word is defined by the recurrence:

S(n) = S(n − 1) ⋅ S(n − 2), for n ≥ 2

with initial seeds S(0) = 0 and S(1) = 01. The length of S(n) satisfies:

|S(n)| = F(n + 2)

where F(k) is the k-th Fibonacci number. The frequency ratio of character "0" converges to the inverse of the golden ratio:

lim_n→∞ |S(n)|₀|S(n)| = 1φ = φ − 1 ≈ 0.6180339887

where φ = 1 + √52 ≈ 1.6180339887. The Fibonacci word can also be generated by the morphism σ: 0 → 01, 1 → 0, iterated from 0.

where S = Fibonacci word string, n = order (generation index), F(k) = k-th Fibonacci number, φ = golden ratio, |S|₀ = count of character "0" in S.

Reference Data

Order n	Length = F(n+2)	Count of "0"	Count of "1"	Density of "0"	Density − 1/φ
0	1	1	0	1.000000	+0.381966
1	2	1	1	0.500000	−0.118034
2	3	2	1	0.666667	+0.048633
3	5	3	2	0.600000	−0.018034
4	8	5	3	0.625000	+0.006966
5	13	8	5	0.615385	−0.002650
6	21	13	8	0.619048	+0.001014
7	34	21	13	0.617647	−0.000387
8	55	34	21	0.618182	+0.000148
9	89	55	34	0.617978	−0.000056
10	144	89	55	0.618056	+0.000022
15	1,597	987	610	0.618034	<0.000001
20	17,711	10,946	6,765	0.618034	<0.000001
25	196,418	121,393	75,025	0.618034	<0.000001
30	2,178,309	1,346,269	832,040	0.618034	<0.000001
35	24,157,817	14,930,352	9,227,465	0.618034	<0.000001
40	267,914,296	165,580,141	102,334,155	0.618034	<0.000001

Frequently Asked Questions

The Fibonacci number sequence adds integers: F(n) = F(n−1) + F(n−2). The Fibonacci word concatenates strings: S(n) = S(n−1) · S(n−2). The length of the n-th Fibonacci word equals the (n+2)-th Fibonacci number, so the two are structurally linked but operate in different domains - arithmetic vs. combinatorics on words.

Each concatenation appends S(n−2) to S(n−1). The count of "0" in S(n) equals the count in S(n−1) plus the count in S(n−2), following the same Fibonacci recurrence as the total length. The ratio of consecutive Fibonacci numbers converges to φ, so the ratio of "0" count to total length converges to F(n+1)/F(n+2) → 1/φ ≈ 0.6180.

Swapping operands produces a different string family. While the lengths remain identical (addition is commutative), the internal structure changes entirely. The resulting words lose the specific subword complexity and palindromic properties of the classical Fibonacci word. Pattern matching results and fractal analysis would be invalid.

Yes. This tool supports custom seeds. The concatenation recurrence S(n) = S(n−1) · S(n−2) applies regardless of seed content. However, the convergence properties - density approaching 1/φ - only hold for the ratio of characters inherited from each seed, weighted by the Fibonacci growth rate. Custom seeds create generalized Fibonacci strings used in certain coding theory applications.

At order 25, the word exceeds 196,000 characters. At order 30, it surpasses 2 million. This tool computes full metadata (length, character counts, density) up to order 40 (267 million characters) but truncates the displayed string at 10,000 characters. The mathematical analysis remains exact regardless of display truncation.

The Fibonacci word is a Sturmian word - it has the lowest subword complexity among non-periodic infinite words, with exactly n+1 distinct factors of length n. The Thue-Morse sequence is cube-free and has different complexity. Both are morphic (generated by iterated substitution), but they belong to different classes in combinatorics on words.