Lucas sequence

A family of integer sequences satisfying the recurrence aₙ = aₙ₋₁ + aₙ₋₂ with different initial values. The classical Lucas numbers begin 2, 1, 3, 4, 7.

Fibonacci recurrence: The rule that each term is the sum of the two preceding terms. It defines the shared step that generates every Lucas sequence.
Initial condition: A specified starting value that selects a solution of a recurrence or differential equation. Changing the first two terms changes the resulting Lucas sequence.
Primality test: An algorithm that determines whether an integer is prime. Special Lucas sequences underpin efficient tests for primality.
Lucas sequence of the first kind: A sequence defined by U₀ = 0, U₁ = 1 and Uₙ = P Uₙ₋₁ − Q Uₙ₋₂. This parameterized family generalizes Fibonacci numbers and is distinct from Lucas numbers.
Characteristic polynomial: A polynomial whose roots encode solutions to a linear recurrence. Its roots produce closed forms for sequences obeying the Fibonacci recurrence.
Fibonacci sequence: The integer sequence beginning 0, 1 in which each later term is the sum of the previous two. It obeys the same recurrence but uses different initial values from the classical Lucas numbers.
Lucas primality test: A primality criterion that uses properties of Lucas sequences modulo an integer. It applies recurrence identities to certify that a candidate number is prime.
Lucas sequence of the second kind: A sequence defined by V₀ = 2, V₁ = P and Vₙ = P Vₙ₋₁ − Q Vₙ₋₂. This family generalizes the classical Lucas numbers using two recurrence parameters.
Binet's formula: A closed-form expression for Fibonacci numbers using powers of the golden ratio and its conjugate. The same root-based method gives closed forms for Lucas numbers and related sequences.
Lucas numbers: The sequence 2, 1, 3, 4, 7, 11, … defined by the Fibonacci recurrence. These are the best-known specific member of the Lucas sequence family.

Linked from 7 pages