Lucas number

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.

The Lucas spiral, made with quarter-arcs, is a good approximation of the golden spiral when its terms are large. However, when its terms become very small, the arc's radius decreases rapidly from 3 to 1 then increases from 1 to 2.

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.[1] 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.[2] 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.[3]

The first few Lucas numbers are

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ....

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.[4] 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 the first twelve Lucas numbers is:

${\displaystyle 2,\;1,\;3,\;4,\;7,\;11,\;18,\;29,\;47,\;76,\;123,\;199,\;\ldots \;}$(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 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 ${\displaystyle L_{n}}$ 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 first identity expressed visually

The Lucas numbers are related to the Fibonacci numbers by many identities. Among these are the following:

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

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 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. These are the Fibonacci pseudoprimes.

L_n - L_n-4 is congruent to 0 mod 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, ... (sequence A005479 in the OEIS).

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, ... (sequence A001606 in the OEIS).

If L_n is prime then n is 0, prime, or a power of 2.[5] L_2^m 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 L_n(x) are a polynomial sequence derived from the Lucas numbers.

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

• Generalizations of Fibonacci numbers

References

1. Weisstein, Eric W. "Lucas Number". mathworld.wolfram.com. Retrieved 2020-08-11.
2. Parker, Matt (2014). "13". Things to Make and Do in the Fourth Dimension. Farrar, Straus and Giroux. p. 284. ISBN 978-0-374-53563-6.
3. Parker, Matt (2014). "13". Things to Make and Do in the Fourth Dimension. Farrar, Straus and Giroux. p. 282. ISBN 978-0-374-53563-6.
4. A New Kind of Science
5. Chris Caldwell, "The Prime Glossary: Lucas prime" from The Prime Pages.

External links

• "Lucas polynomials", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Lucas Number". MathWorld.
• Weisstein, Eric W. "Lucas Polynomial". MathWorld.
• "The Lucas Numbers", Dr Ron Knott
• Lucas numbers and the Golden Section
• A Lucas Number Calculator can be found here.
• OEIS sequence A000032 (Lucas numbers beginning at 2)