Lucas number

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Not to be confused with Lucas sequences, a generic class of sequences to which the Lucas numbers belong.

The Lucas numbers or Lucas series are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–91), who studied both that sequence and the closely related Fibonacci numbers. Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequences.

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:

${\displaystyle L_{n}:={\begin{cases}2&{\text{if }}n=0;\\1&{\text{if }}n=1;\\L_{n-1}+L_{n-2}&{\text{if }}n>1.\\\end{cases}}}$

(where n belongs to the natural numbers)

The sequence of Lucas numbers is:

${\displaystyle 1,\;3,\;4,\;7,\;11,\;18,\;29,\;47,\;76,\;123,\;\ldots \;}$(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 Ln−2 = Ln − Ln−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 ${\displaystyle L_{n}}$ for ${\displaystyle -5\leq {}n\leq 5}$ are shown).

The formula for terms with negative indices in this sequence is

${\displaystyle L_{-n}=(-1)^{n}L_{n}.\!}$

Relationship to Fibonacci numbers

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

• ${\displaystyle \,L_{n}=F_{n-1}+F_{n+1}=F_{n}+2F_{n-1}=F_{n+2}-F_{n-2}}$
• ${\displaystyle \,L_{m+n}=L_{m+1}F_{n}+L_{m}F_{n-1}}$
• ${\displaystyle \,L_{n}^{2}=5F_{n}^{2}+4(-1)^{n}}$, and thus as ${\displaystyle n\,}$ approaches +∞, the ratio ${\displaystyle {\frac {L_{n}}{F_{n}}}}$ approaches ${\displaystyle {\sqrt {5}}.}$
• ${\displaystyle \,F_{2n}=L_{n}F_{n}}$
• ${\displaystyle \,F_{n+k}+(-1)^{k}F_{n-k}=L_{k}F_{n}}$
• ${\displaystyle \,F_{n}={L_{n-1}+L_{n+1} \over 5}={L_{n-3}+L_{n+3} \over 10}}$

Their closed formula is given as:

${\displaystyle L_{n}=\varphi ^{n}+(1-\varphi )^{-n}=\varphi ^{n}+(-\varphi )^{-n}=\left({1+{\sqrt {5}} \over 2}\right)^{n}+\left({1-{\sqrt {5}} \over 2}\right)^{n}\,,}$

where ${\displaystyle \varphi }$ is the golden ratio. Alternatively, as for ${\displaystyle n>1}$ the magnitude of the term ${\displaystyle (-\varphi )^{-n}}$ is less than 1/2, ${\displaystyle L_{n}}$ is the closest integer to ${\displaystyle \varphi ^{n}}$ or, equivalently, the integer part of ${\displaystyle \varphi ^{n}+1/2}$, also written as ${\displaystyle \lfloor \varphi ^{n}+1/2\rfloor }$.

Combining the above with Binet's formula,

${\displaystyle F_{n}={\frac {\varphi ^{n}-(1-\varphi )^{n}}{\sqrt {5}}}\,,}$

a formula for ${\displaystyle \varphi ^{n}}$ is obtained:

${\displaystyle \varphi ^{n}={{L_{n}+F_{n}{\sqrt {5}}} \over 2}\,.}$

Congruence relations

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

Ln 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 Ln is prime then n is either 0, prime, or a power of 2.[1] L2m is prime for m = 1, 2, 3, and 4 and no other known values of m.

Generating series

Let

${\displaystyle \Phi (x)=2+x+3x^{2}+4x^{3}+\cdots =\sum _{n=0}^{\infty }L_{n}x^{n}}$

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

{\displaystyle {\begin{aligned}\Phi (x)&=L_{0}+L_{1}x+\sum _{n=2}^{\infty }L_{n}x^{n}\\&=2+x+\sum _{n=2}^{\infty }(L_{n-1}+L_{n-2})x^{n}\\&=2+x+\sum _{n=1}^{\infty }L_{n}x^{n+1}+\sum _{n=0}^{\infty }L_{n}x^{n+2}\\&=2+x+x(\Phi (x)-2)+x^{2}\Phi (x)\end{aligned}}}

which can be rearranged as

${\displaystyle \Phi (x)={\frac {2-x}{1-x-x^{2}}}.}$

The partial fraction decomposition is given by

${\displaystyle \Phi (x)={\frac {1}{1-\varphi x}}+{\frac {1}{1-\phi x}}}$

where ${\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}}$ is the golden ratio and ${\displaystyle \phi ={\frac {1-{\sqrt {5}}}{2}}}$ is its conjugate.

Lucas polynomials

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