Infinite integer series where the next number is the sum of the two preceding it
Not to be confused with Lucas sequences, the generic class of sequences to which the Lucas numbers belong.
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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 numbers or Lucas series are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci numbers. Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequences.
The Lucas series 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]
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 and as opposed to the first two Fibonacci numbers and . 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 the first twelve 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.
Extension to negative integers[]
Using , 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 for are shown).
The formula for terms with negative indices in this sequence is
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:
, so .
; in particular, , so .
Their closed formula is given as:
where is the golden ratio. Alternatively, as for the magnitude of the term is less than 1/2, is the closest integer to or, equivalently, the integer part of , also written as .
As of September 2015, the largest confirmed Lucas prime is L148091, which has 30950 decimal digits.[4] As of June 2017, the largest known Lucas probable prime is L2316773, with 484177 decimal digits.[5]
If Ln is prime then n is 0, prime, or a power of 2.[6]L2m is prime for m = 1, 2, 3, and 4 and no other known values of m.
Lucas polynomials[]
In the same way as Fibonacci polynomials are derived from the Fibonacci numbers, the Lucas polynomials 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[]
^Weisstein, Eric W. "Lucas Number". mathworld.wolfram.com. Retrieved 2020-08-11.
^Parker, Matt (2014). "13". Things to Make and Do in the Fourth Dimension. Farrar, Straus and Giroux. p. 284. ISBN978-0-374-53563-6.
^Parker, Matt (2014). "13". Things to Make and Do in the Fourth Dimension. Farrar, Straus and Giroux. p. 282. ISBN978-0-374-53563-6.