400x240px|thumb|right|The Lucas spiral, made with quarter-[[circular arc|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 is an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), 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

: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, ... .

which coincides for example with the number of independent vertex sets for cyclic graphs <math>C_n</math> of length <math>n\geq2</math>. , the largest known Lucas probable prime is L<sub>5466311</sub>, with 1,142,392 decimal digits.

If L<sub>n</sub> is prime then n is 0, prime, or a power of 2. L<sub>2<sup>m</sup></sub> is prime for m&nbsp;= 1,&nbsp;2,&nbsp;3,&nbsp;and&nbsp;4 and no other known values of&nbsp;m.

Lucas polynomials

In the same way as Fibonacci polynomials are derived from the Fibonacci numbers, the Lucas polynomials <math>L_{n}(x)</math> 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&nbsp; very closely approximates the -th power of the golden ratio,&nbsp;. 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:

:<math> \varphi^{2n-1} = [L_{2n-1}; L_{2n-1}, L_{2n-1}, L_{2n-1}, \ldots] </math>

:<math> \varphi^{2n} = [L_{2n}-1; 1, L_{2n}-2, 1, L_{2n}-2, 1, L_{2n}-2, 1, \ldots] </math>.

For example:

:<math> \varphi^5 = [11; 11, 11, 11, \ldots] </math>

is the limit of

:<math> \frac{11}{1}, \frac{122}{11}, \frac{1353}{122}, \frac{15005}{1353}, \ldots </math>

with the error in each term being about 1% of the error in the previous term; and

:<math> \varphi^6 = [18 - 1; 1, 18 - 2, 1, 18 - 2, 1, 18 - 2, 1, \ldots] = [17; 1, 16, 1, 16, 1, 16, 1, \ldots] </math>

is the limit of

:<math> \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 </math>

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

  • Generalizations of Fibonacci numbers

References

  • "The Lucas Numbers", Dr Ron Knott
  • Lucas numbers and the Golden Section
  • A Lucas Number Calculator can be found here.

bn:লুকাস ধারা

fr:Suite de Lucas

he:סדרת לוקאס

pt:Sequência de Lucas