Talk:Lucas number

Comments on proposed merger
Comments should be on Talk:Lucas sequence, at the proposed target article which the merge box links to on "Discuss". Duplicating here is not necessary. PrimeHunter 18:12, 21 January 2007 (UTC)

I disagree with the proposed merger of Lucas number into Lucas sequence. Lucas sequences encompass not just Lucas numbers but also Fibonacci numbers, Pell numbers, and in fact any sequence defined by a linear recurrence relation with a quadratic characteristic equation. Making Lucas numbers a special case by merging them into the Lucas sequence article would be anomalous and misleading. Gandalf61 09:42, 21 January 2007 (UTC)

Do not merge the articles. Lucas numbers deserve their own article just as much as Fibonacci numbers do. And Lucas sequences not only encompass a broad range of famous integer sequences; they're also useful in applied mathematics (pseudo-primality testing, etc). DavidCBryant 12:46, 21 January 2007 (UTC)


 * Against merger This article is about a specific important sequence. It is simple and easy to understand. The other article is very abstract and difficult to understand and is about many sequences, some of which may not be so important as this one. However, we should have a link to it as a generalization. In other words: Keep It Simple, Stupid! JRSpriggs 13:32, 21 January 2007 (UTC)

what this article also needs is a formula that can be used to calculate the nth term in the lucas numbers sequence!!!!!! —Preceding unsigned comment added by 121.222.188.10 (talk) 02:18, 9 May 2010 (UTC)

There should not be credit given to lucas for realizing that just because you start somewhere else the aplied formula is still the applied formula and does not change or add to the integral knowledge that all here inlies One sequence of of liner occurance in relation to a quadratic equation, im disagreeing with Gandalf61, lucas is nothing special out of fibbonacci numbers all lucas preposals are credited out of context User:Xinbone —Preceding undated comment added 02:36, 25 May 2016 (UTC)

Lucas numbers generated by Pascal Triangle analogue
In the Wiki article on the Pascal Triangle there is a subsection showing how the Fibonacci numbers can be generated by summing samplings taken across every other diagonal. I've seen, online (but can't remember URL), an analogue to the Pascal triangle with one of the 1's sides replaced by 2's. Repeating the aforementioned procedure on this modified Pascal system gives the Lucas numbers in one direction, and the Fibonacci in the other. In fact any Fibonacci-like sequence, leading to the Golden Ratio, can be created by adjusting the sides of the triangle, and the 'seeds' of these sequences are in fact the numbers on the sides. So for the classical Pascal Triangle, with 1,1 on the sides, gives (1,1),2,3,5,8... and you can see that ..(1,2),3,5,8.. leads to a similar, but offset Fib. In the other direction (2,1),3,4,7,11... you get Lucas. This may seem like original research (it was for me) but I have to believe that mathematicians have known about this for ages, and there must be published sources one could hang the donkey tail onto so as to be able to include these facts into the main articles. Anyone know of any? Thanks. — Preceding unsigned comment added by 96.234.78.93 (talk) 11:28, 13 October 2011 (UTC)

Links
The link "A Tutorial on Generalized Lucas Numbers" is no longer valid. Anybody knows a new location? — Preceding unsigned comment added by 217.5.143.100 (talk) 07:55, 11 June 2014 (UTC)


 * I haven't found a live version. It's archived at https://web.archive.org/web/20130825030815/http://nakedprogrammer.com/LucasNumbers.aspx and http://www.archive.today/LBOG1 PrimeHunter (talk) 09:09, 11 June 2014 (UTC)

External links modified
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 * Added archive https://web.archive.org/web/20051030021553/http://milan.milanovic.org/math/english/lucas/lucas.html to http://milan.milanovic.org/math/english/lucas/lucas.html
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Relationship to Fibonacci numbers addition
Hereby I want to send some generalisation formulas for two Items in the chapter Relationship to Fibonacci numbers.

The item $$\,L_n = F_{n-1}+F_{n+1}=F_n+2F_{n-1} = F_{n+2}-F_{n-2}$$ can be more generalise by the following equation:

L_n := \begin{cases} {F_{n-m}+F_{n+m} \over F_{m}}          & \text{if m is uneven and } m => 1;\\ {F_{n-m}-F_{n+m} \over F_{m}}          & \text{if m is even and } m => 2\\ \end{cases} $$

where m is de position away from the Lucas number to find.

and item $$\,F_n = {L_{n-1}+L_{n+1} \over 5} = {L_{n-3}+L_{n+3} \over 10} $$ can be generalised with the following equations



F_n := \begin{cases} {L_{n+m} + L_{n-m} \over {5F_{m}}}          & \text{if m is uneven and } m => 1;\\ {L_{n+m} - L_{n-m} \over {5F_{m}}}          & \text{if m is even and } m => 2\\ \end{cases} $$

which we can rewrite as



F_nF_m := \begin{cases} {L_{n+m} + L_{n-m} \over 5}          & \text{if m is uneven and } m => 1;\\ {L_{n+m} - L_{n-m} \over 5}          & \text{if m is even and } m => 2\\ \end{cases} $$

or only written as Lucas numbers



F_n := \begin{cases} {L_{n+m} + L_{n-m} \over {L_{m+1} + L_{m-1}}}          & \text{if m is uneven and } m => 1;\\ {L_{n+m} - L_{n-m} \over {L_{m+1} + L_{m-1}}}          & \text{if m is even and } m => 2\\ \end{cases} $$

— Preceding unsigned comment added by 86.84.102.126 (talk) 06:31, 22 September 2018 (UTC)

Useful identities
Next identities are very usefull if you need to simplify some complex fibs expressions. Besides, they are elegant. --Ivigan (talk) 06:33, 5 July 2019 (UTC)
 * $$F_{n+k} + (-1)^k F_{n-k} = L_k F_n, \quad F_{n+k} - (-1)^k F_{n-k} = L_n F_k$$
 * $$L_{n+k} + (-1)^k L_{n-k} = L_k L_n, \quad L_{n+k} - (-1)^k L_{n-k} = 5 F_n F_k$$
 * I have edited your comment for readability. Three of these identities are in the article already (the second is the same as the first).  The other one does not involve Fibonacci numbers, but you placed it in the section titled "Relationship to Fibonacci numbers".  It can also easily be derived from formulas that are in the article.  --JBL (talk) 10:14, 5 July 2019 (UTC)