Lucas sequences
E337572
Lucas sequences are a family of integer sequences defined by the same type of second-order linear recurrence as the Fibonacci numbers but with more general initial conditions, encompassing the Fibonacci sequence as a special case.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Jacobsthal numbers | 1 |
| Lucas number sequence | 1 |
| Lucas sequences canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3214184 — 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.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Lucas sequences Context triple: [Fibonacci sequence, isSubsequenceOf, Lucas sequences]
-
A.
Gaussian periods
Gaussian periods are special algebraic sums of roots of unity that play a key role in number theory, particularly in constructing regular polygons like the 17-gon with straightedge and compass and in understanding cyclotomic fields.
-
B.
Bernoulli numbers
Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
-
C.
Ulam sequence
The Ulam sequence is an integer sequence starting with 1 and 2 in which each subsequent term is the smallest integer that can be written uniquely as the sum of two distinct earlier terms.
-
D.
Berlekamp–Massey algorithm
The Berlekamp–Massey algorithm is a key algorithm in coding theory and cryptography used to efficiently determine the shortest linear feedback shift register that generates a given binary sequence.
-
E.
Chebyshev functions
Chebyshev functions are arithmetic functions in number theory that encode information about the distribution of prime numbers and play a key role in analytic approaches to the prime number theorem.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Lucas sequences Target entity description: Lucas sequences are a family of integer sequences defined by the same type of second-order linear recurrence as the Fibonacci numbers but with more general initial conditions, encompassing the Fibonacci sequence as a special case.
-
A.
Gaussian periods
Gaussian periods are special algebraic sums of roots of unity that play a key role in number theory, particularly in constructing regular polygons like the 17-gon with straightedge and compass and in understanding cyclotomic fields.
-
B.
Bernoulli numbers
Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
-
C.
Ulam sequence
The Ulam sequence is an integer sequence starting with 1 and 2 in which each subsequent term is the smallest integer that can be written uniquely as the sum of two distinct earlier terms.
-
D.
Berlekamp–Massey algorithm
The Berlekamp–Massey algorithm is a key algorithm in coding theory and cryptography used to efficiently determine the shortest linear feedback shift register that generates a given binary sequence.
-
E.
Chebyshev functions
Chebyshev functions are arithmetic functions in number theory that encode information about the distribution of prime numbers and play a key role in analytic approaches to the prime number theorem.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
integer sequence family
ⓘ
mathematical concept ⓘ number theory concept ⓘ |
| appearsIn |
combinatorics
ⓘ
recurrence relation theory ⓘ |
| classification | second-order linear homogeneous recurrence sequences ⓘ |
| constraint | P and Q are usually integers ⓘ |
| definedBy | second-order linear recurrence relation ⓘ |
| dependOn | P and Q ⓘ |
| field |
discrete mathematics
ⓘ
number theory ⓘ |
| firstKindInitialConditions | U_0 = 0, U_1 = 1 ⓘ |
| generalizes |
Fibonacci sequence
ⓘ
Lucas number sequence ⓘ
linked to:
Lucas sequences
|
| hasApplication |
cryptography via primality tests
ⓘ
pseudorandom number generation ⓘ |
| hasNotation |
U_n(P,Q)
ⓘ
V_n(P,Q) ⓘ |
| hasOnlineResource | OEIS entries for many specific Lucas sequences ⓘ |
| hasSubfamily |
Lucas sequence of the first kind
ⓘ
Lucas sequence of the second kind ⓘ |
| includesSequence |
Fibonacci numbers
ⓘ
linked to:
Fibonacci sequence
Jacobsthal numbers ⓘ
linked to:
Lucas sequences
Lucas numbers ⓘ Pell numbers ⓘ |
| namedAfter | Édouard Lucas ⓘ |
| parameter |
P
ⓘ
Q ⓘ |
| property |
satisfy linear recurrence with constant coefficients
ⓘ
terms are integers for integer P and Q ⓘ |
| recurrenceType | u_n = P u_{n-1} - Q u_{n-2} ⓘ |
| relatedConcept | Lucas pair (P,Q) ⓘ |
| relatedTo |
Binet-type closed forms
ⓘ
characteristic polynomial x^2 - Px + Q ⓘ linear recurrences modulo m ⓘ roots of x^2 - Px + Q ⓘ |
| secondKindInitialConditions | V_0 = 2, V_1 = P ⓘ |
| specialCaseCondition |
Fibonacci sequence corresponds to P = 1, Q = -1 with suitable initial conditions
ⓘ
linked to:
Fibonacci sequence
Lucas numbers correspond to P = 1, Q = -1 with different initial conditions ⓘ |
| termIndex | n ⓘ |
| usedIn |
Diophantine equations
ⓘ
Lucas probable prime tests ⓘ Lucas test ⓘ Lucas–Lehmer test ⓘ algebraic number theory ⓘ primality testing ⓘ |
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.
Instruction
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.
Input
Subject: Lucas sequences Description of subject: Lucas sequences are a family of integer sequences defined by the same type of second-order linear recurrence as the Fibonacci numbers but with more general initial conditions, encompassing the Fibonacci sequence as a special case.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Lucas sequences
linked to: Lucas sequences