Fibonacci sequence
E67350
The Fibonacci sequence is an infinite series of numbers where each term is the sum of the two preceding ones, widely used in mathematics, art, and design due to its connection with the golden ratio and natural growth patterns.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Fibonacci numbers | 10 |
| Fibonacci sequence canonical | 9 |
| Binet's formula | 1 |
| Binet’s formula | 1 |
| Fibonacci sequence corresponds to P = 1, Q = -1 with suitable initial conditions | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T538848 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
Target entity: Fibonacci sequence Context triple: [Le Modulor, basedOn, Fibonacci sequence]
-
A.
Look-and-say sequence
The look-and-say sequence is a famous integer sequence where each term is generated by verbally describing the digits of the previous term, studied for its surprising combinatorial and growth properties.
-
B.
Pascal's triangle
Pascal's triangle is a triangular array of numbers in which each entry is the sum of the two directly above it, widely used in combinatorics, algebra, and probability.
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C.
Euler’s totient function φ(n)
Euler’s totient function φ(n) is a fundamental arithmetic function in number theory that counts the positive integers up to n that are relatively prime to n and plays a key role in topics such as modular arithmetic and cryptography.
-
D.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
-
E.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
Target entity: Fibonacci sequence Target entity description: The Fibonacci sequence is an infinite series of numbers where each term is the sum of the two preceding ones, widely used in mathematics, art, and design due to its connection with the golden ratio and natural growth patterns.
-
A.
Look-and-say sequence
The look-and-say sequence is a famous integer sequence where each term is generated by verbally describing the digits of the previous term, studied for its surprising combinatorial and growth properties.
-
B.
Pascal's triangle
Pascal's triangle is a triangular array of numbers in which each entry is the sum of the two directly above it, widely used in combinatorics, algebra, and probability.
-
C.
Euler’s totient function φ(n)
Euler’s totient function φ(n) is a fundamental arithmetic function in number theory that counts the positive integers up to n that are relatively prime to n and plays a key role in topics such as modular arithmetic and cryptography.
-
D.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
-
E.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
- F. None of above. chosen
Statements (53)
| Predicate | Object |
|---|---|
| instanceOf |
integer sequence
ⓘ
mathematical concept ⓘ |
| alternativeName |
Fibonacci sequence
ⓘ
surface form:
Fibonacci numbers
|
| appearsIn |
phyllotaxis patterns
ⓘ
pinecone spirals ⓘ rabbit population model ⓘ spiral arrangements in plants ⓘ sunflower seed patterns ⓘ |
| asymptoticForm | F(n) ≈ φ^n / √5 ⓘ |
| belongsTo |
combinatorics
ⓘ
discrete mathematics ⓘ number theory ⓘ |
| definedByRecurrence | F(n) = F(n−1) + F(n−2) ⓘ |
| firstTerms |
0
ⓘ
1 ⓘ 13 ⓘ 2 ⓘ 21 ⓘ 3 ⓘ 34 ⓘ 5 ⓘ 55 ⓘ 8 ⓘ |
| hasCharacteristicPolynomial | x^2 − x − 1 ⓘ |
| hasClosedForm |
Fibonacci sequence
self-linksurface differs
ⓘ
surface form:
Binet’s formula
|
| hasFirstTerm | 0 ⓘ |
| hasGeneratingFunction | x / (1 − x − x^2) ⓘ |
| hasGrowthRate | exponential ⓘ |
| hasInitialConditions |
F(0) = 0
ⓘ
F(1) = 1 ⓘ |
| hasParityPatternPeriod | 3 ⓘ |
| hasSecondTerm | 1 ⓘ |
| isDefinedOn | nonnegative integers ⓘ |
| isExampleOf |
linear homogeneous recurrence with constant coefficients
ⓘ
recursively defined sequence ⓘ |
| isImplementedIn | many programming tutorials ⓘ |
| isInfinite | true ⓘ |
| isLinearRecurrenceOfOrder | 2 ⓘ |
| isStrictlyIncreasingFromIndex | 2 ⓘ |
| isSubsequenceOf | Lucas sequences ⓘ |
| isUsedInAlgorithm |
Fibonacci heap
ⓘ
Fibonacci search ⓘ dynamic programming examples ⓘ |
| limitOfRatioOfConsecutiveTerms | golden ratio ⓘ |
| moduloPatternName | Pisano period ⓘ |
| namedAfter | Leonardo Fibonacci ⓘ |
| relatedTo | golden ratio ⓘ |
| usedIn |
architecture
ⓘ
art ⓘ computer algorithms ⓘ design ⓘ mathematical modeling ⓘ nature modeling ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Subject: Fibonacci sequence Description of subject: The Fibonacci sequence is an infinite series of numbers where each term is the sum of the two preceding ones, widely used in mathematics, art, and design due to its connection with the golden ratio and natural growth patterns.
Referenced by (22)
Full triples — surface form annotated when it differs from this entity's canonical label.