Lucas number

Definition

Similar to the Fibonacci numbers, each Lucas number is defined to be the sum of its two immediate previous terms, thereby forming a Fibonacci integer sequence. The first two Lucas numbers are L₀ = 2 and L₁ = 1 as opposed to the first two Fibonacci numbers F₀ = 0 and F₁ = 1. Though closely related in definition, Lucas and Fibonacci numbers exhibit distinct properties.

The Lucas numbers may thus be defined as follows:

L n := { 2 if  n = 0 ; 1 if  n = 1 ; L n − 1 + L n − 2 if  n > 1.

(where n belongs to the natural numbers)

The sequence of Lucas numbers is:

1 , 3 , 4 , 7 , 11 , 18 , 29 , 47 , 76 , 123 , … (sequence A000032 in the OEIS).

All Fibonacci-like integer sequences appear in shifted form as a row of the Wythoff array; the Fibonacci sequence itself is the first row and the Lucas sequence is the second row. Also like all Fibonacci-like integer sequences, the ratio between two consecutive Lucas numbers converges to the golden ratio.

Extension to negative integers

Using L_n−2 = L_n − L_n−1, one can extend the Lucas numbers to negative integers to obtain a doubly infinite sequence:

..., −11, 7, −4, 3, −1, 2, 1, 3, 4, 7, 11, ... (terms L n for − 5 ≤ n ≤ 5 are shown).

The formula for terms with negative indices in this sequence is

L − n = ( − 1 ) n L n .

Relationship to Fibonacci numbers

The Lucas numbers are related to the Fibonacci numbers by the identities

L n = F n − 1 + F n + 1 = F n + 2 F n − 1 = F n + 2 − F n − 2

L m + n = L m + 1 F n + L m F n − 1

L n 2 = 5 F n 2 + 4 ( − 1 ) n , and thus as n approaches +∞, the ratio L n F n approaches 5 .

F 2 n = L n F n

F n + k + ( − 1 ) k F n − k = L k F n

F n = L n − 1 + L n + 1 5 = L n − 3 + L n + 3 10

Their closed formula is given as:

L n = φ n + ( 1 − φ ) n = φ n + ( − φ ) − n = ( 1 + 5 2 ) n + ( 1 − 5 2 ) n ,

where φ is the golden ratio. Alternatively, as for n > 1 the magnitude of the term ( − φ ) − n is less than 1/2, L n is the closest integer to φ n or, equivalently, the integer part of φ n + 1 / 2 , also written as ⌊ φ n + 1 / 2 ⌋ .

Combining the above with Binet's formula,

F n = φ n − ( 1 − φ ) n 5 ,

a formula for φ n is obtained:

φ n = L n + F n 5 2 .

Congruence relations

If F_n ≥ 5 is a Fibonacci number then no Lucas number is divisible by F_n.

L_n is congruent to 1 mod n if n is prime, but some composite values of n also have this property.

Lucas primes

A Lucas prime is a Lucas number that is prime. The first few Lucas primes are

2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, ... (sequence A005479 in the OEIS).

For these ns are

0, 2, 4, 5, 7, 8, 11, 13, 16, 17, 19, 31, 37, 41, 47, 53, 61, 71, 79, 113, 313, 353, 503, 613, 617, 863, 1097, 1361, 4787, 4793, 5851, 7741, 8467, ... (sequence A001606 in the OEIS).

If L_n is prime then n is either 0, prime, or a power of 2. L_2^m is prime for m = 1, 2, 3, and 4 and no other known values of m.

Generating series

Let

Φ ( x ) = 2 + x + 3 x 2 + 4 x 3 + ⋯ = ∑ n = 0 ∞ L n x n

be the generating series of the Lucas numbers. By a direct computation,

Φ ( x ) = L 0 + L 1 x + ∑ n = 2 ∞ L n x n = 2 + x + ∑ n = 2 ∞ ( L n − 1 + L n − 2 ) x n = 2 + x + ∑ n = 1 ∞ L n x n + 1 + ∑ n = 0 ∞ L n x n + 2 = 2 + x + x ( Φ ( x ) − 2 ) + x 2 Φ ( x )

which can be rearranged as

Φ ( x ) = 2 − x 1 − x − x 2 .

The partial fraction decomposition is given by

Φ ( x ) = 1 1 − φ x + 1 1 − ϕ x

where φ = 1 + 5 2 is the golden ratio and ϕ = 1 − 5 2 is its conjugate.

Lucas polynomials

In the same way as Fibonacci polynomials are derived from the Fibonacci numbers, the Lucas polynomials L_n(x) are a polynomial sequence derived from the Lucas numbers.