Lucas number

As with the Fibonacci numbers, each Lucas number is defined to be the sum of its two immediately previous terms, thereby forming a Fibonacci integer sequence. The first two Lucas numbers are L_0=2 and L_1=1, which differs from the first two Fibonacci numbers F_0=0 and F_1=1. Though closely related in definition, Lucas and Fibonacci numbers exhibit distinct properties. The Lucas numbers may thus be defined as follows: : L_n := \begin{cases} 2 & \text{if } n = 0; \\ 1 & \text{if } n = 1; \\ L_{n-1}+L_{n-2} & \text{if } n > 1. \end{cases} (where n belongs to the natural numbers) 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\leq{}n\leq5 are shown). The formula for terms with negative indices in this sequence is : L_{-n}=(-1)^nL_n.\! ==Relationship to Fibonacci numbers==

The Lucas numbers are related to the Fibonacci numbers by many identities. Among these are the following: • L_n = F_{n+1}+F_{n-1} = F_{n+2}-F_{n-2}. • L_{n+k} = L_{n+1}F_k + L_n F_{k-1}. • F_{n+k} = L_k F_n - (-1)^k F_{n-k}; in particular when k=n, F_{2n} = L_n F_n. • 2F_{n+k} = L_n F_k + L_k F_n. • L_{2n} = 5 F_n^2 + 2(-1)^n = L_n^2 - 2(-1)^n, so \lim_{n\to\infty} \frac{L_n}{F_n}=\sqrt{5}. • L_n - \sqrt{5} F_n = \frac{2}{(-\varphi)^n} \to 0. • L_{n+k} - (-1)^k L_{n-k} = 5 F_n F_k; in particular when k=1, 5F_n = L_{n+1}+L_{n-1} = 2L_{n+1} - L_n = L_n + 2L_{n-1}, while when k=n, the identity above for L_{2n} is obtained. Their closed formula is given as: : L_n = \varphi^n + (1-\varphi)^{n} = \varphi^n + (-\varphi)^{-n} = \left({ 1+ \sqrt{5} \over 2}\right)^n + \left({ 1- \sqrt{5} \over 2}\right)^n , where \varphi is the golden ratio. For n>1 the magnitude of the term (1-\varphi)^n is less than 1/2, therefore L_n is the closest integer to \varphi^n or, equivalently, the integer part of \varphi^n+1/2, also written as \lfloor \varphi^n+1/2 \rfloor. Combining the above with Binet's formula, :F_n = \frac{\varphi^n - (1-\varphi)^n}{\sqrt 5} = \frac{\varphi^n - (-\varphi)^{-n}}{\sqrt 5} \, , a formula for \varphi^n is obtained: :\varphi^n = {L_n + F_n \sqrt 5 \over 2}\, . For integers n ≥ 2, we also get: : \varphi^n = L_n - (1-\varphi)^n = L_n - (-1)^n L_n^{-1} - L_n^{-3} + R with remainder R satisfying : \vert R \vert ==Lucas identities==

Many of the Fibonacci identities have parallels in Lucas numbers. For example, the Cassini identity becomes :L_n^2 - L_{n-1}L_{n+1} = (-1)^{n}5 Also :\sum_{k=0}^n L_k = L_{n+2} - 1 :\sum_{k=0}^n L_k^2 = L_nL_{n+1} + 2 :2L_{n-1}^2 + L_n^2 = L_{2n+1} + 5F_{n-2}^2 where \textstyle F_n=\frac{L_{n-1}+L_{n+1}}{5}. : L_n^k = \sum_{j=0}^{\lfloor \frac{k}{2} \rfloor} (-1)^{nj} \binom{k}{j} L'_{(k-2j)n} where L'_n=L_n except for L'_0=1. For example if n is odd, L_n^3 = L'_{3n}-3L'_n and L_n^4 = L'_{4n}-4L'_{2n}+6L'_0 Checking, L_3=4, 4^3=64=76-3(4), and 256=322-4(18)+6 ==Generating function==

The ordinary generating function of the sequence of Lucas numbers is the power series \Phi(x) = \sum_{k=0}^\infty L_k x^k = 2 + x + 3x^2 + 4x^3 + 7x^4 + 11x^5 + \cdots. This series is convergent for any complex number x satisfying |x| and its sum has a simple closed form: \Phi(x)=\frac{2-x}{1-x-x^2}. This can be proved by multiplying by (1-x-x^2): \begin{align} (1 - x- x^2) \Phi(x) &= \sum_{k=0}^{\infty} L_k x^k - \sum_{k=0}^{\infty} L_k x^{k+1} - \sum_{k=0}^{\infty} L_k x^{k+2} \\ &= \sum_{k=0}^{\infty} L_k x^k - \sum_{k=1}^{\infty} L_{k-1} x^k - \sum_{k=2}^{\infty} L_{k-2} x^k \\ &= 2x^0 + 1x^1 - 2x^1 + \sum_{k=2}^{\infty} (L_k - L_{k-1} - L_{k-2}) x^k \\ &= 2 - x, \end{align} where all terms involving x^k for k \ge 2 cancel out because of the defining Lucas numbers recurrence relation. \Phi\!\left(-\frac{1}{x}\right) gives the generating function for the negative indexed Lucas numbers, \sum_{n = 0}^\infty (-1)^nL_nx^{-n} = \sum_{n = 0}^\infty L_{-n}x^{-n}, and :\Phi\!\left(-\frac{1}{x}\right) = \frac{x + 2x^2}{1 - x - x^2} \Phi(x) satisfies the functional equation :\Phi(x) - \Phi\!\left(-\frac{1}{x}\right) = 2 As the generating function for the Fibonacci numbers is given by :s(x) = \frac{x}{1 - x - x^2} we have :s(x) + \Phi(x) = \frac{2}{1 - x - x^2} which proves that :F_n + L_n = 2F_{n+1}, and :5s(x) + \Phi(x) = \frac2x\Phi(-\frac1x) = 2\frac{1}{1 - x - x^2} + 4\frac{x}{1 - x - x^2} proves that :5F_n + L_n = 2L_{n+1} The partial fraction decomposition is given by :\Phi(x) = \frac{1}{1 - \phi x} + \frac{1}{1 - \psi x} where \phi = \frac{1 + \sqrt{5}}{2} is the golden ratio and \psi = \frac{1 - \sqrt{5}}{2} is its conjugate. This can be used to prove the generating function, as :\sum_{n = 0}^\infty L_nx^n = \sum_{n = 0}^\infty (\phi^n + \psi^n)x^n = \sum_{n = 0}^\infty \phi^nx^n + \sum_{n = 0}^\infty \psi^nx^n = \frac{1}{1 - \phi x} + \frac{1}{1 - \psi x} = \Phi(x) Using x equal to any of 0.01, 0.001, 0.0001, etc. lays out the first Lucas numbers in the decimal expansion of \Phi(x). For example, \Phi(0.001) = \frac{1.999}{0.998999} = \frac{1999000}{998999} = 2.001003004007011018029047\ldots. ==Congruence relations==

If F_n\geq 5 is a Fibonacci number then no Lucas number is divisible by F_n. The Lucas numbers satisfy Gauss congruence. This implies that L_n is congruent to 1 modulo n if n is prime. The composite values of n which satisfy this property are known as Fibonacci pseudoprimes. L_n-L_{n-4} is congruent to 0 modulo 5. == 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, ... . The indices of these primes are (for example, L4 = 7) :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, ... . , 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 L_{n}(x) 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: : \varphi^{2n-1} = [L_{2n-1}; L_{2n-1}, L_{2n-1}, L_{2n-1}, \ldots] : \varphi^{2n} = [L_{2n}-1; 1, L_{2n}-2, 1, L_{2n}-2, 1, L_{2n}-2, 1, \ldots] . For example: : \varphi^5 = [11; 11, 11, 11, \ldots] is the limit of : \frac{11}{1}, \frac{122}{11}, \frac{1353}{122}, \frac{15005}{1353}, \ldots with the error in each term being about 1% of the error in the previous term; and : \varphi^6 = [18 - 1; 1, 18 - 2, 1, 18 - 2, 1, 18 - 2, 1, \ldots] = [17; 1, 16, 1, 16, 1, 16, 1, \ldots] is the limit of : \frac{17}{1}, \frac{18}{1}, \frac{305}{17}, \frac{323}{18}, \frac{5473}{305}, \frac{5796}{323}, \frac{98209}{5473}, \frac{104005}{5796}, \ldots with the error in each term being about 0.3% that of the second previous term. ==Applications==

Lucas numbers are the second most common pattern in sunflowers after Fibonacci numbers, when clockwise and counter-clockwise spirals are counted, according to an analysis of 657 sunflowers in 2016. ==See also==