Editing Lucas number (section)

{{short description|Infinite integer series where the next number is the sum of the two preceding it}}
{{distinguish|text=[[Lucas sequence]]s, the general class of sequences to which the Lucas numbers belong}}
{{More footnotes|date=December 2019}}
[[File:Lucas number spiral.svg|400x240px|thumb|right|The Lucas spiral, made with quarter-[[circular arc|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 sequence''' is an [[integer sequence]] named after the mathematician [[Édouard Lucas|François Édouard Anatole Lucas]] (1842&ndash;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 sequence]]s. 

The Lucas sequence has the same [[recurrence relation|recursive relationship]] as the Fibonacci sequence, where each term is the sum of the two previous terms, but with different starting values.<ref name="mathworld wolfram weisstein">{{Cite web|last=Weisstein|first=Eric W.|title=Lucas Number|url=https://mathworld.wolfram.com/LucasNumber.html|access-date=2020-08-11|website=mathworld.wolfram.com|language=en}}</ref> This produces a sequence where the ratios of successive terms approach the [[golden ratio]], and in fact the terms themselves are [[rounding]]s of [[integer]] powers of the golden ratio.<ref>{{cite book |last1=Parker |first1=Matt |title=Things to Make and Do in the Fourth Dimension |date=2014 |publisher=Farrar, Straus and Giroux |isbn=978-0-374-53563-6 |page=284 |language=English |chapter=13}}</ref> 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.<ref>{{cite book |last1=Parker |first1=Matt |title=Things to Make and Do in the Fourth Dimension |date=2014 |publisher=Farrar, Straus and Giroux |isbn=978-0-374-53563-6 |page=282 |language=English |chapter=13}}</ref>

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, ...&thinsp;. {{OEIS|id=A000032}}

which coincides for example with the number of [[Independent set (graph theory)|independent vertex sets ]] for [[cyclic graph|cyclic graphs]] <math>C_n</math> of length <math>n\geq2</math>.<ref name="mathworld wolfram weisstein" />