Editing Lucas sequence (section)

== Specific names ==

The Lucas sequences for some values of ''P'' and ''Q'' have specific names:

:{{math|''U<sub>n</sub>''(1, −1)}} : [[Fibonacci number]]s
:{{math|''V<sub>n</sub>''(1, −1)}} : [[Lucas number]]s
:{{math|''U<sub>n</sub>''(2, −1)}} : [[Pell number]]s
:{{math|''V<sub>n</sub>''(2, −1)}} : [[Pell–Lucas numbers]] (companion Pell numbers)
:{{math|''U<sub>n</sub>''(1, −2)}} : [[Jacobsthal number]]s
:{{math|''V<sub>n</sub>''(1, −2)}} : [[Jacobsthal–Lucas numbers]]
:{{math|''U<sub>n</sub>''(3, 2)}} : [[Mersenne number]]s 2<sup>''n''</sup>&nbsp;−&thinsp;1
:{{math|''V<sub>n</sub>''(3, 2)}} : Numbers of the form 2<sup>''n''</sup>&nbsp;+&thinsp;1, which include the [[Fermat number]]s<ref name=Yabuta/>
:{{math|''U<sub>n</sub>''(6,&thinsp;1)}} : The square roots of the [[square triangular number]]s.
:{{math|''U<sub>n</sub>''(''x'', −1)}} : [[Fibonacci polynomials]]
:{{math|''V<sub>n</sub>''(''x'', −1)}} : [[Lucas polynomials]]
:{{math|''U<sub>n</sub>''(2''x'',&thinsp;1)}} : [[Chebyshev polynomials]] of second kind
:{{math|''V<sub>n</sub>''(2''x'',&thinsp;1)}} : [[Chebyshev polynomials]] of first kind multiplied by 2
:{{math|''U<sub>n</sub>''(''x''+1, ''x'')}} : [[Repunit]]s in base ''x''
:{{math|''V<sub>n</sub>''(''x''+1, ''x'')}} : ''x<sup>n</sup>''&nbsp;+&thinsp;1

Some Lucas sequences have entries in the [[On-Line Encyclopedia of Integer Sequences]]:

:{|class="wikitable" style="background: #fff"
|-
!<math>P\,</math>!!<math>Q\, </math>!!<math>U_n(P,Q)\, </math>!! <math>V_n(P,Q)\,</math>
|- 
| −1 || 3 || {{OEIS2C|A214733}}
|-
| 1 || −1 || {{OEIS2C|A000045}} || {{OEIS2C|A000032}}
|-
| 1 || 1 || {{OEIS2C|A128834}} || {{OEIS2C|A087204}}
|-
| 1 || 2 || {{OEIS2C|A107920}} || {{OEIS2C|A002249}}
|-
| 2 || −1 || {{OEIS2C|A000129}} || {{OEIS2C|A002203}}
|-
| 2 || 1 || {{OEIS2C|A001477}} || {{OEIS2C|A007395}}
|-
| 2 || 2 || {{OEIS2C|A009545}} 
|-
| 2 || 3 || {{OEIS2C|A088137}}
|-
| 2 || 4 || {{OEIS2C|A088138}}
|-
| 2 || 5 || {{OEIS2C|A045873}}
|-
| 3 || −5 || {{OEIS2C|A015523}} || {{OEIS2C|A072263}}
|-
| 3 || −4 || {{OEIS2C|A015521}} || {{OEIS2C|A201455}}
|-
| 3 || −3 || {{OEIS2C|A030195}} || {{OEIS2C|A172012}}
|-
| 3 || −2 || {{OEIS2C|A007482}} || {{OEIS2C|A206776}}
|-
| 3 || −1 || {{OEIS2C|A006190}} || {{OEIS2C|A006497}}
|-
| 3 || 1 || {{OEIS2C|A001906}} || {{OEIS2C|A005248}}
|-
| 3 || 2 || {{OEIS2C|A000225}} || {{OEIS2C|A000051}}
|-
| 3 || 5 || {{OEIS2C|A190959}}
|-
| 4 || −3 || {{OEIS2C|A015530}} || {{OEIS2C|A080042}}
|-
| 4 || −2 || {{OEIS2C|A090017}}
|-
| 4 || −1 || {{OEIS2C|A001076}} || {{OEIS2C|A014448}}
|-
| 4 || 1 || {{OEIS2C|A001353}} || {{OEIS2C|A003500}}
|-
| 4 || 2 || {{OEIS2C|A007070}}  || {{OEIS2C|A056236}}
|-
| 4 || 3 || {{OEIS2C|A003462}} || {{OEIS2C|A034472}}
|-
| 4 || 4 || {{OEIS2C|A001787}}
|-
| 5 || −3 || {{OEIS2C|A015536}}
|-
| 5 || −2 || {{OEIS2C|A015535}}
|-
| 5 || −1 || {{OEIS2C|A052918}} || {{OEIS2C|A087130}}
|-
| 5 || 1 || {{OEIS2C|A004254}} || {{OEIS2C|A003501}}
|-
| 5 || 4 ||{{OEIS2C|A002450}} || {{OEIS2C|A052539}}
|-
| 6 || 1 ||{{OEIS2C|A001109}} || {{OEIS2C|A003499}}
|}