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== Fibonacci numbers vs. Fibonacci sequence ==

We bothered to change the name of the article from [[Fibonacci numbers]] to [[Fibonacci sequence]].  However, much of the text still says things like "The Fibonacci numbers are" rather than "The Fibonacci sequence is".  I realize that there are some instances where we really do mean the former, and I realize that there are some instances where changing the former to the latter would be a word salad, and I realize that we don't have to be pedantic about every occurrence ... but might it be worthwhile to change many of instances of the "Fibonacci numbers" to "Fibonacci sequence"?  Or, putting it another way, if I do that, am I likely to get instantly reverted?  Thanks —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 14:39, 2 February 2024 (UTC)

:No big deal indeed, so no problem with me. Afaiac, go ahead {{smiley}}. - [[User:DVdm|DVdm]] ([[User talk:DVdm|talk]]) 15:41, 2 February 2024 (UTC)
:However, the article title and your personal preferences are not good reasons for the change, and, per [[MOS:VAR]], you must provide stronger reasons.
:Also, the two phrases are not always equivalent. For example, in the last but one paragraph of the lead, the examples given are related to the first numbers of the sequence only, not to the whole sequence. So, I would oppose strongly to the change in this paragraph. In the last paragraph of the lead, this is different, as the Fibonacci numbers are not individually related to the golden ratio; this is the sequence that is related to it. So, in this case, I would strongly support the change. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 16:30, 2 February 2024 (UTC)
:: I agree. Of course we can only make changes where it makes sense. - [[User:DVdm|DVdm]] ([[User talk:DVdm|talk]]) 16:41, 2 February 2024 (UTC)
::{{tq|i=yes|Fibonacci numbers are not individually related to the golden ratio}}
::This claim seems too broad and pretty pedantic. For example, powers of the golden ratio when written as "golden integers" of the form <math>a + b\varphi</math> have "individual Fibonacci numbers" as their coefficients, as described in {{slink|Fibonacci sequence#Decomposition of powers}}. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 17:30, 2 February 2024 (UTC)
:::Of course, I would not have used this sentence in the article. However, to establish the expression of the powers of the golden ratio, one needs the recurrence relation, and thus the defition of the sequence. The fact is that it is better to use "sequence" when all numbers are considered together. So "sequence" is better in the first sentence (before the colon) and the last sentence of the last paragraph of the lead; "numbers" is better in the remainder of this paragraph. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 17:55, 2 February 2024 (UTC)
:Did no one notice that Fibonacci numbers and Fibonacci sequence mean the exact same thing, or is it just me? [[Special:Contributions/80.42.238.212|80.42.238.212]] ([[User talk:80.42.238.212|talk]]) 11:26, 8 July 2024 (UTC)
::"Sequence" implies an order among the numbers ... a first Fibonacci number, then a second, then a third, ....  However, "numbers" need not be so organized.  For example, among the transcendental numbers, there is none that people agree is the first, then the second, etc.  —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 12:37, 8 July 2024 (UTC)
:::Thanks; I guess that kind of helps. I only thought that because when I hovered over Fibonacci numbers and then over Fibonacci sequences, it gave the same definition. [[Special:Contributions/80.42.238.212|80.42.238.212]] ([[User talk:80.42.238.212|talk]]) 08:40, 10 July 2024 (UTC)
::The distinction between "Fibonacci numbers" and "Fibonacci sequence" is pedantry.  No one who says "Fibonacci numbers" means a random set of numbers; they always mean the standard increasing sequence of numbers, unless they are talking about individual members of the sequence.  Changing the article title seems silly to me; after all, look up "[[Lucas numbers]]" for comparison. [[User:Zaslav|Zaslav]] ([[User talk:Zaslav|talk]]) 05:01, 12 October 2024 (UTC)
:::Well said! [[User:Imaginatorium|Imaginatorium]] ([[User talk:Imaginatorium|talk]]) 05:53, 12 October 2024 (UTC)

== Legibility ==

With [https://en.wikipedia.org/w/index.php?title=Fibonacci_sequence&oldid=prev&diff=1240299192 this edit] to the last example in section [[Fibonacci sequence#Fibonacci primes]], for better legibility, I changed this:
<math display=block>F_1 = 1,\ F_3 = 2,\ F_5 = 5,\ F_7 = 13,\ F_9 = 34 = 2 \cdot 17,\ F_{11} = 89,\ F_{13} = 233,\ F_{15} = 610 = 2 \cdot 5 \cdot 61.</math>
into this:
<math display=block>\begin{align}
F_1 &= 1\\
F_3 &= 2\\
F_5 &= 5\\
F_7 &= 13\\
F_9 &= 34 = 2 \cdot 17\\
F_{11} &= 89\\
F_{13} &= 233\\
F_{15} &= 610 = 2 \cdot 5 \cdot 61
\end{align}</math>
User {{u|JayBeeEll}} reverted ([https://en.wikipedia.org/w/index.php?title=Fibonacci_sequence&diff=next&oldid=1240281973]) with reason "''I disagree that this increases legibility overall''". Obviously I disagree with JBL's disagreement {{smiley}} because I think that my version does a much better job at highlighting the primes. What do others think about this? - [[User:DVdm|DVdm]] ([[User talk:DVdm|talk]]) 17:37, 14 August 2024 (UTC)

:I don't really care either way.  But we could instead highlight the primes in the more compact form by using color. —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 20:38, 20 August 2024 (UTC)
::Yes, that's another way to make 'em stand out a bit more than they do now. - [[User:DVdm|DVdm]] ([[User talk:DVdm|talk]]) 21:02, 20 August 2024 (UTC)
:Thanks for starting a discussion here, sorry for my cryptic comment and for the delayed response. The substance of my objection is that making this vertical devotes a very large amount of space in the article to a very minor point, which seemed to me to distract from everything else.  I would be happy with color as an alternative, or with a two-line array (say F_1 through F_9 on the first line, F_11 through F_15 on the second) that would allow one to easily pick out the factorization in the two composites. --[[User:JayBeeEll|JBL]] ([[User_talk:JayBeeEll|talk]]) 00:36, 21 August 2024 (UTC)
::Good point. Let's go for the color then. I went ahead and colored the ''non''-primes that could be factored, leaving 1 black: [https://en.wikipedia.org/w/index.php?title=Fibonacci_sequence&diff=1241468242&oldid=1241423241]. 
::<math display=block>F_1 = 1,\ F_3 = 2,\ F_5 = 5,\ F_7 = 13,\ F_9 = {\color{Red}34} = 2 \cdot 17,\ F_{11} = 89,\ F_{13} = 233,\ F_{15} = {\color{Red}610} = 2 \cdot 5 \cdot 61.</math>
::Cheers! = [[User:DVdm|DVdm]] ([[User talk:DVdm|talk]]) 10:04, 21 August 2024 (UTC)

== They don't all have closed forms ==

The article claims "Like every [[sequence]] defined by a [[linear recurrence with constant coefficients]], the Fibonacci numbers have a [[closed-form expression]]", but this is not true. It is true for all linear recurrences of order 4 or lower, and for some special cases of higher orders. But it is certainly not true for all of them. Perhaps moderate that sentence a bit, or maybe even remove the claim entirely? (Keeping the link to [[linear recurrence with constant coefficients]] seems apt, though.) [[User:MasterHigure|MasterHigure]] ([[User talk:MasterHigure|talk]]) 20:18, 20 August 2024 (UTC)

:The definition of "closed-form expression" in the first sentence of the article [[Closed-form expression]] allows the use of "constants", in this case the roots of the characteristic polynomial.  When the coefficients of the recurrence are rational, if the degree is larger than 4, these roots may not be expressible in terms of radicals; but that seems irrelevant.  --[[User:JayBeeEll|JBL]] ([[User_talk:JayBeeEll|talk]]) 20:27, 20 August 2024 (UTC)
:Perhaps relevant: a quintic equation (and likewise for higher degree) is not guaranteed to give a closed form for its roots.  But if we are handed the roots, we could write down a closed form expression for the sequence.  That is, there is a closed form expression for the sequence, it's just that generally we can't figure it out.  —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 20:34, 20 August 2024 (UTC)

:{{ec}}
:This source
::* {{cite book |title=Discrete Mathematics with Ducks |author1=Sarah-Marie Belcastro |edition=2nd, illustrated |publisher=CRC Press |year=2018 |isbn=978-1-351-68369-2 |page=260 |url=https://books.google.com/books?id=xoqADwAAQBAJ}} [https://books.google.com/books?id=xoqADwAAQBAJ&pg=PA260 Extract of page 260]
: says... {{xt|...there is an algorithm for finding a closed form for any linear '''homogenous''' recurrence relation with constant coefficients.}}
: SInce the Fibonacci expression is indeed homogenous, I have just added the word ''homogenous'' to the sentence and added the source. So we don't have to worry about the order. - [[User:DVdm|DVdm]] ([[User talk:DVdm|talk]]) 20:49, 20 August 2024 (UTC)
::"Homogenous" is not a word in English.  I fixed it. [[User:Zaslav|Zaslav]] ([[User talk:Zaslav|talk]]) 20:19, 7 October 2024 (UTC)
::: {{rto|Zaslav}} Homogenous is an English word. See https://www.onelook.com/?w=Homogenous
::: So I restored the original spelling per [[wp:ENVAR]] and [[wp:RETAIN]]. - [[User:DVdm|DVdm]] ([[User talk:DVdm|talk]]) 20:49, 7 October 2024 (UTC)
:::I have to agree with @[[User:Zaslav|Zaslav]]; the cited source (if reproduced correctly) has a misspelling; it should be "homogeneous".  ([https://www.grammar.com/homogenous_vs._homogeneous This page] says that the version without an "e" is obsolete.) —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 20:53, 7 October 2024 (UTC)
::::I also agree. To me, homogenous is a perfectly good word, but Google Scholar shows that for this technical meaning homogeneous is overwhelmingly preferred. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 20:55, 7 October 2024 (UTC)
::::: Ok, I agree to follow the cited source. Thanks for your comments. - [[User:DVdm|DVdm]] ([[User talk:DVdm|talk]]) 20:57, 7 October 2024 (UTC)
::::::According to the latest research (2 minutes ago on line), "homogenous" may be a technical term in biology (possibly replaced by "homologous").  As a variant of "homogeneous" in good English, Merriam-Webster says, 
:::::::Homogeneous comes from the Greek roots hom-, meaning "same," and genos, meaning "kind." The similar word homogenous is a synonym of the same origin.  
::::::My primary source for the incorrectness of "homogenous" as a synonym is [[Anna Russell]], who in one of her famous routines (I think, but am not sure, it was the 22-minute ''Ring'' cycle) said, "I mean homogenous, as in milk!" [[User:Zaslav|Zaslav]] ([[User talk:Zaslav|talk]]) 23:12, 7 October 2024 (UTC)
:::::::That was exactly the response of a colleague who teaches lower-division discrete mathematics (where this term appears) whom I asked about the name for this type of recurrence. He responded "homogeneous", and when I asked about "homogenous" he responded something about whether I meant the word for milk. I think that is usually "homogenized", though. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 00:38, 8 October 2024 (UTC)
::::::::Your colleague was pulling your leg.  You have to listen to [https://www.youtube.com/watch?v=aYCXO_FZj5k Anna Russell] to understand this.  I quote from a Web site about [https://tvtropes.org/pmwiki/pmwiki.php/Creator/AnnaRussell "How to Write Your Own Gilbert and Sullivan Opera"]:
::::::::"As you know, you always have to start with a homogenous chorus. I know a lot of people are going to say that isn't homogenous, that's homogeneous. But that isn't what I mean: I mean homogenous, as in milk." [[User:Zaslav|Zaslav]] ([[User talk:Zaslav|talk]]) 05:10, 12 October 2024 (UTC)
:::::::::I do not understand why this discussion continue. It is clear that "homogeneous" is a correct English word (see above quotation of Merriam-Webster) and it is clear that, in mathematics, the standard word is homogeneous. If you are not convinced, search Scholar Google with "homogenous recurrence" and "homogenous equation": you will find ''no result'' with the asked spelling and more than 6,000,000 with the spelling "homogeneous". [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 09:45, 12 October 2024 (UTC)

== Complementary? ==

The intro says "Lucas numbers ... with the Fibonacci numbers form a complementary pair of Lucas sequences."  This concept is not defined anywhere in this article, in [[Lucas sequence]], or in  [[Lucas number]].  [[User:Zaslav|Zaslav]] ([[User talk:Zaslav|talk]]) 20:16, 7 October 2024 (UTC)

:See [[Lucas sequence#Specific names]]. The word "complementary pair" is not defined in the linked article, but is seems clear from the context. Feel free to add an explicit definition near the definition of "first kind" and "second kind". [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 10:05, 12 October 2024 (UTC)
::I find this terminology confusing, because it conflicts with the meaning of complementary in e.g. [[Lambek–Moser theorem]], where sequences are complementary when every positive integer belongs to exactly one of them. Moreover, its explanation cannot be found in [[Lucas sequence]]. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 17:51, 12 October 2024 (UTC)
:::I agree. However, if there are reliable sources showing that "complementary" is a standard term here, it must be kept and defined in [[Lucas sequence]]. Otherwise, I suggest to replace it with "associated". [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 07:56, 13 October 2024 (UTC)

== Removing the OEIS number ==

Someone added the OEIS sequence number after the initial introduction of the series. I feel this is a kind of category error - WP is supposed to be an encyclopedia about the real world, not a kind of index to various works of administration. Supremely, the Fibonacci sequence is what it is defined to be, where each term is the sum of the two preceding terms. Everyone agrees on (at least this part of) the definition, and it stands above anyone's attempt to catalogue sequences, however valuable this attempt (OEIS) is. It reminds me of people who think that any character (such as a numeral digit 1) requires a list of all the ways it might be represented in Unicode. [[User:Imaginatorium|Imaginatorium]] ([[User talk:Imaginatorium|talk]]) 02:59, 6 December 2024 (UTC)

:Your second error (after removing this useful link) is to call OEIS a "work of administration". Geographic articles contain latitude/longitude links (usually in a prominent position near the top) linking to more specialized topic-specific references (online mapping systems) for that lat/lon. Think of this the same way: a more specialized topic-specific reference for integer sequences, that readers looking up this sequence would probably also want to consult. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 03:13, 6 December 2024 (UTC)
::But note the already templated citation ''immediately'' preceding the line: ''<nowiki>{{Cite OEIS|1=A000045|2=Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1|mode=cs2}}</ref></nowiki>''.
::Can't we avoid this repetition of links to both [[OEIS]] and [[OEIS:A000045]]? [[User:DVdm|DVdm]] ([[User talk:DVdm|talk]]) 10:11, 6 December 2024 (UTC)
::I second what DVdm says.  We already cite the OEIS sequence right next to the new citation.  Let's keep exactly one of these, yes?  —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 13:40, 6 December 2024 (UTC)
:::I would rather keep the more prominent inline OEIS link and ditch the fatuous lead-reference for how the sequence starts. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 18:47, 6 December 2024 (UTC)
::::That works for me.  Please boldly edit.  —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 18:51, 6 December 2024 (UTC)