Lucas numbers

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.

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:

(where n belongs to the natural numbers)

The sequence of Lucas numbers is:

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.

Using L_n−2 = L_n − L_n−1, one can extend the Lucas numbers to negative integers to obtain a doubly infinite sequence:

The formula for terms with negative indices in this sequence is

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

Their closed formula is given as:

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 ${\displaystyle \lfloor \varphi ^{n}+1/2\rfloor }$.

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