Lucas numbers

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The sequence of Lucas numbers is named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related sequence of Fibonacci numbers (both sequences are Lucas sequences).

The Lucas numbers are defined by the following homogeneous linear recurrence of order

2

and signature

(1, 1)

(see Category:Recurrence, linear, order 02, (1,1))

$L_{0}:=2,\,$

$L_{1}:=1,\,$

$L_{n}:=L_{n-1}+L_{n-2},\quad n\geq 2.\,$

A000032 Lucas numbers (beginning at

2

):

L (0) = 2; L (1) = 1; L (n) = L (n − 1) + L (n − 2), n ≥ 2

. (See A000204 for Lucas numbers beginning with

1

.)

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, 167761, 271443, 439204, 710647, 1149851, 1860498, 3010349, 4870847, ...

The Lucas numbers can be obtained from the Fibonacci numbers thus:

$L_{n}=F_{n-1}+F_{n+1}=2F_{n-1}+F_{n}.\,$

From Binet's closed-form formula for Fibonacci numbers we can readily derive a closed formula for the Lucas numbers:

$L_{n}=\left({\frac {1-{\sqrt {5}}}{2}}\right)^{n}+\left({\frac {1+{\sqrt {5}}}{2}}\right)^{n}$

(the latter summand is a power of the golden ratio $\phi$).

The ratio of consecutive Lucas numbers converges to the same limit as for the Fibonacci numbers, namely the golden ratio:

$\lim _{n\to \infty }{\frac {L_{n+1}}{L_{n}}}=\lim _{n\to \infty }{\frac {F_{n+1}}{F_{n}}}=\phi .\,$

See also

• {{Lucas}} (mathematical function template)