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.[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]
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 and , which differs from the first two Fibonacci numbers and . Though closely related in definition, Lucas and Fibonacci numbers exhibit distinct properties.
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 Lucas numbers are related to the Fibonacci numbers by many identities. Among these are the following:
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[update], the largest confirmed Lucas prime is L148091, which has 30950 decimal digits.[4] As of August 2022[update], the largest known Lucas probable prime is L5466311, with 1,142,392 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.
For positive integers n, the continued fractions are:
.
For example:
is the limit of
with the error in each term being about 1% of the error in the previous term; and
is the limit of
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.[7]
^ a bWeisstein, 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. ISBN 978-0-374-53563-6.
^Parker, Matt (2014). "13". Things to Make and Do in the Fourth Dimension. Farrar, Straus and Giroux. p. 282. ISBN 978-0-374-53563-6.
^"The Top Twenty: Lucas Number". primes.utm.edu. Retrieved 6 January 2022.
^"Henri & Renaud Lifchitz's PRP Top - Search by form". www.primenumbers.net. Retrieved 6 January 2022.
^Chris Caldwell, "The Prime Glossary: Lucas prime" from The Prime Pages.
^Swinton, Jonathan; Ochu, Erinma; null, null (2016). "Novel Fibonacci and non-Fibonacci structure in the sunflower: results of a citizen science experiment". Royal Society Open Science. 3 (5): 160091. Bibcode:2016RSOS....360091S. doi:10.1098/rsos.160091. PMC 4892450. PMID 27293788.