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, ... . (sequence A000032 in the OEIS) which coincides for example with the number of independent vertex sets for cyclic graphs C n {\displaystyle C_{n}} of length n ≥ 2 {\displaystyle n\geq 2} .
Definition 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 {\displaystyle L_{0}=2} and L 1 = 1 {\displaystyle L_{1}=1} , which differs from the first two Fibonacci numbers F 0 = 0 {\displaystyle F_{0}=0} and F 1 = 1 {\displaystyle 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 := { 2 if n = 0 ; 1 if n = 1 ; L n − 1 + L n − 2 if n > 1. {\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) 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 {\displaystyle 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 {\displaystyle L_{n}} for − 5 ≤
n ≤ 5 {\displaystyle -5\leq {}n\leq 5} are shown). The formula for terms with negative indices in this sequence is
L − n = ( − 1 ) n L n . {\displaystyle L_{-n}=(-1)^{n}L_{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 . {\displaystyle L_{n}=F_{n+1}+F_{n-1}=F_{n+2}-F_{n-2}.}
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