Lucas number
The Lucas sequence is an integer sequence named after the mathematician François Édouard Anatole Lucas, who studied both that sequence and the closely related Fibonacci sequence. Individual numbers in the Lucas sequence are known as Lucas numbers. Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequences.
The Lucas sequence has the same recursive relationship as the Fibonacci sequence, where each term is the sum of the two previous terms, but with different starting values. This produces a sequence where the ratios of successive terms approach the golden ratio, and in fact the terms themselves are roundings of integer powers of the golden ratio. The sequence also has a variety of relationships with the Fibonacci numbers, like the fact that adding any two Fibonacci numbers two terms apart in the Fibonacci sequence results in the Lucas number in between.
The first few Lucas numbers are
which coincides for example with the number of independent vertex sets for cyclic graphs of length.
Definition
As with the Fibonacci numbers, each Lucas number is defined to be the sum of its two immediately previous terms, thereby forming a Generalizations of Fibonacci numbers#Fibonacci [integer sequences|Fibonacci integer sequence]. The first two Lucas numbers are and, which differs from the first two Fibonacci numbers and. Though closely related in definition, Lucas and Fibonacci numbers exhibit distinct properties.The Lucas numbers may thus be defined as follows:
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, one can extend the Lucas numbers to negative integers to obtain a doubly infinite sequence:The formula for terms with negative indices in this sequence is
Relationship to Fibonacci numbers
The Lucas numbers are related to the Fibonacci numbers by many identities. Among these are the following:- , so.
- ; in particular,, so.
where is the golden ratio. Alternatively, as for the magnitude of the term is less than 1/2, is the closest integer to or, equivalently, the integer part of, also written as.
Combining the above with Binet's formula,
a formula for is obtained:
For integers n ≥ 2, we also get:
with remainder R satisfying
Lucas identities
Many of the Fibonacci identities have parallels in Lucas numbers. For example, the Cassini identity becomesAlso
where.
where except for.
For example if n is odd, and
Checking,, and
Generating function
The ordinary generating function of the sequence of Lucas numbers is the power seriesThis series is convergent for any complex number satisfying and its sum has a simple closed form:
This can be proved by multiplying by :
where all terms involving for cancel out because of the defining Lucas numbers recurrence relation.
gives the generating function for the negative indexed Lucas numbers,, and
satisfies the functional equation
As the generating function for the Fibonacci numbers is given by
we have
which proves that
and
proves that
The partial fraction decomposition is given by
where is the golden ratio and is its conjugate.
This can be used to prove the generating function, as
Using equal to any of 0.01, 0.001, 0.0001, etc. lays out the first Lucas numbers in the decimal expansion of. For example,
Congruence relations
If is a Fibonacci number then no Lucas number is divisible by.The Lucas numbers satisfy Gauss congruence. This implies that is congruent to 1 modulo if is prime. The composite values of which satisfy this property are known as Fibonacci pseudoprimes.
is congruent to 0 modulo 5.
Lucas primes
A Lucas prime is a Lucas number that is prime. The first few Lucas primes areThe indices of these primes are
, the largest confirmed Lucas prime is L148091, which has 30950 decimal digits., the largest known Lucas probable prime is L5466311, with 1,142,392 decimal digits.
If Ln is prime then n is 0, prime, or a power of 2. L2m is prime for m = 1, 2, 3, and 4 and no other known values of m.
Lucas polynomials
In the same way as Fibonacci polynomials are derived from the Fibonacci numbers, the Lucas polynomials are a polynomial sequence derived from the Lucas numbers.Continued fractions for powers of the golden ratio
For all but the smallest values of, the integer very closely approximates the -th power of the golden ratio, . Furthermore, close rational approximations for powers of the golden ratio can be obtained from their continued fractions.For positive integers n, the continued fractions are:
For example:
is the limit of
with the error in each term being about 1% of the error in the previous term; and
is the limit of
with the error in each term being about 0.3% that of the second previous term.