Talk:Fibonacci sequence

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.) 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. --JBL (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. —Quantling (talk | contribs) 20:34, 20 August 2024 (UTC)

(edit conflict)

This source

• Sarah-Marie Belcastro (2018). Discrete Mathematics with Ducks (2nd, illustrated ed.). CRC Press. p. 260. ISBN 978-1-351-68369-2. Extract of page 260

says... ...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. - DVdm (talk) 20:49, 20 August 2024 (UTC)

"Homogenous" is not a word in English. I fixed it. Zaslav (talk) 20:19, 7 October 2024 (UTC)

@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. - DVdm (talk) 20:49, 7 October 2024 (UTC)

I have to agree with @Zaslav; the cited source (if reproduced correctly) has a misspelling; it should be "homogeneous". (This page says that the version without an "e" is obsolete.) —Quantling (talk | 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. —David Eppstein (talk) 20:55, 7 October 2024 (UTC)

Ok, I agree to follow the cited source. Thanks for your comments. - 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!" 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. —David Eppstein (talk) 00:38, 8 October 2024 (UTC)

Your colleague was pulling your leg. You have to listen to Anna Russell to understand this. I quote from a Web site about "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." 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". 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. 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". 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. —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". 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. 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. —David Eppstein (talk) 03:13, 6 December 2024 (UTC)

But note the already templated citation immediately preceding the line: {{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>.

Can't we avoid this repetition of links to both OEIS and OEIS:A000045? 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? —Quantling (talk | 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. —David Eppstein (talk) 18:47, 6 December 2024 (UTC)

That works for me. Please boldly edit. —Quantling (talk | contribs) 18:51, 6 December 2024 (UTC)

Fibonacci identities

The article contains several identities on finding specific values using the smaller values, however I feel like it lacks some of the easier identities such as the powers of two and the sum of two values (on a technicality it contains this, yet there is a simpler formula), I would like to add these identities however I am unable to find a source to cite them, I was wondering if I should make a separate category for them as I cannot cite sources or prove them even though they work, should I add these extra identities or leave it be? 206.186.188.206 (talk) 00:35, 12 July 2025 (UTC)

If you cannot find a source, you should not add them. If you can find a single obscure source, you might still not want to add them. If you can find them mentioned in a widely cited book or survey paper, then please go for it. If you need research help, you can try asking here or at Wikipedia:Reference desk/Mathematics. (If you came up with a true identity, it's almost certain to have been published before. Among other places, there are 60 years of issues of Fibonacci Quarterly full of such identities.) –jacobolus (t) 00:41, 12 July 2025 (UTC)

Well I have been unable to find sources stating F(a+b)=F(a+1)F(b)+F(a)F(b-1) and even when I asked people in a math discord server I was told that it didn’t seem to be documented, which is strange as it comes directly from the definition of Fibonacci numbers and also shows d’Ocagne’s identity for F(2n) when you plug in a=b=n, and my other identities stem from this, as for my other identity it just generalizes d’Ocagne’s identity to all powers of two, you can see it is true by just looking at the original identity and doing it recursively 206.186.188.206 (talk) 01:04, 12 July 2025 (UTC)

Page 88 of doi:10.1002/9781118033067.ch5. –jacobolus (t) 01:48, 12 July 2025 (UTC)

Also see:

Honsberger, Ross (1985). "A Second Look at the Fibonacci and Lucas Numbers". Mathematical Gems III. Washington, DC: Mathematical Association of America. Ch. 8.

–jacobolus (t) 02:29, 12 July 2025 (UTC)

Ah, okay, would these sources be enough? You need to pay to see the one 206.186.188.206 (talk) 15:27, 12 July 2025 (UTC)

ldots or cdots

@Jacobolus et al., is there a Wikipedia manual of style for using ldots or cdots in mathematical expressions? I was taught to use ldots if the first omitted item sits on the baseline (or like the letter "y", falls below it) and to use cdots if the first omitted thing sits above it, such as most binary operations.

So, I'd write: $$s(z)=\sum _{k=0}^{\infty }F_{k}z^{k}=0+z+z^{2}+2z^{3}+3z^{4}+5z^{5}+\ldots .$$ or, if I felt the need to omit the addition sign too, then I'd write $$s(z)=\sum _{k=0}^{\infty }F_{k}z^{k}=0+z+z^{2}+2z^{3}+3z^{4}+5z^{5}\cdots .$$ (Admittedly, the rule of thumb gets a little fuzzy with implied operations (like ab being equivalent to a·b in many contexts, so is it ⁠$a\ldots$⁠ or ⁠$a\cdots$⁠?) and with binary operations that don't sit above the baseline.)

What rule / guideline are you using? —Quantling (talk | contribs) 12:21, 15 July 2025 (UTC)

There's another example on the same page with centered dots (in § Applications » Mathematics). My impression is that the usual practice in mathematical typesetting nowadays follows "The dots should appear at the baseline when between commas, and in the center of the line when between other operators" (found here from a quick web search). –jacobolus (t) 15:19, 15 July 2025 (UTC)

Thank you for the quick response. I still like the rule I was taught better, but it seems there are alternatives out there, so ⁠$\cdots$⁠ whatever. Thanks —Quantling (talk | contribs) 16:27, 15 July 2025 (UTC)

I think pre-Latex, most typesetting systems only had one type of ellipsis, so historical practice was often to just use "ldots" for everything. But using centered dots for arbitrary operators seems to be the general best practice in recent times for mathematical writing in English. For example, the SIAM style guide (p. 65) says: "Line dots (ldots) are used between variables with other punctuation. Centered dots (cdots) are used between operators." Wiley's mathematical typesetting advice (I won't link it as it's a word doc, lol) says "In elided sums or elided relations, the ellipsis points should be vertically centered between the operation or relation signs". Feel free to search for other style guides / typesetting guides if you want.

It would probably be even better to use semantically meaningful \dotsc (⁠$\dotsc$⁠) for commas, \dotsb (⁠$\dotsb$⁠) for binary operations, \dotsm (⁠$\dotsm$⁠) for multiplication by juxtaposition, \dotsi (⁠$\dotsi$⁠) for integrals, or \dotso (⁠$\dotso$⁠) for others. In theory \dots is supposed to figure out which one to use from the context, but the heuristic isn't super sophisticated and it often gets it wrong as in this case. When it goes wrong I usually just write \ldots or \cdots. –jacobolus (t) 17:43, 15 July 2025 (UTC)

By the way, you shouldn't "omit the addition sign". That's usually considered incorrect. –jacobolus (t) 17:49, 15 July 2025 (UTC)

I find it interesting that my way puts the dots at a level where the omitted text would have been and your way puts the dots at a level where the immediately adjacent (not-omitted) text is. For example, I would write a set of binary operators as ⁠$\{+,-,\times ,\cdots \}$⁠ rather than ⁠$\{+,-,\times ,\ldots \}$⁠.

I agree that generally it would be awkward to omit the addition sign immediately before the ellipsis of a series. If we're looking for a contrived exception... suppose for each term you get to flip a coin to decide whether the next term is positive or negative. The the series might end up looking like ⁠${}-1-x+x^{2}+x^{3}-x^{4}+x^{5}-x^{6}\cdots$⁠. Yes, I warned you it was contrived. Would you also use cdots there? —Quantling (talk | contribs) 18:42, 15 July 2025 (UTC)

In your contrived example the correct thing to do is either $-x^{6}\pm \cdots$ or $-x^{6}\pm \ldots$. Personally I tend to use ldots when the operation is addition, but \cdots looks a lot better following \pm for some reason. --JBL (talk) 00:47, 16 July 2025 (UTC)

In your example, you should not omit the operation. As JBL says, use ⁠$\pm$⁠. In your other example, I'd go for the lower dots when using commas, even if the entries are centered. I have never seen this come up though. Anyway, if you want to consistently use 'ldots' on some page or other, that's probably fine; there are some authors who do that, and according to Wikipedia (with questionable sourcing) it's standard in Russian mathematical typesetting. –jacobolus (t) 04:06, 16 July 2025 (UTC)

It case it isn't clear ... I wouldn't ever escalate this to an edit war or any formal dispute, because I know that my approach is only one of several. I continue this discussion only because I find the topic interesting that there are these multiple ways. And maybe I can learn something.

Because I'm contriving the example, I could make it a three-sided coin where the next term is either added, subtracted, or omitted. Or it could be addition vs. subtraction vs. a binary operation * signifying some other binary operation under discussion. Maybe then we'd put the ellipsis (centered or lowered) starting after a term rather than after a binary operator. Or maybe you have syntax that generalizes ⁠$\pm$⁠ that would handle that too!

But even with just addition and subtraction... conceptually the random choices being made are to extend the sequence with ⁠${}-x^{4}$⁠, then ⁠${}+x^{5}$⁠, then ⁠${}-x^{6}$⁠, etc. so it seems wrong to me to place the ellipsis into the middle of one of those extending steps.

But that's all contrived stuff. The part that actually interests me: If we expand to include areas outside of mathematics, is there a general rule for how high to put the ellipses? In this more general context I wonder how my preference to put the ellipses at the level roughly where the start of the omitted text would be balances with the preference to put the ellipsis at the level where the last of the previous not-omitted text is. Perhaps the answer is boring (at least for English): is it that only in mathematics do we ever use anything other than ldots? —Quantling (talk | contribs) 14:14, 16 July 2025 (UTC)

FWIW, ChatGPT disagrees with me: Does the level that the ellipsis is placed depend upon the omitted text's level or the level of the surrounding text? ... the placement of an ellipsis depends primarily on the level and type of the surrounding text—not on the content that is being omitted. —Quantling (talk | contribs) 15:22, 16 July 2025 (UTC)

Please don't try to use ChatGPT to answer research questions. It is literally a stochastic bullshit generator. (Feel free to use ChatGPT to write bad poetry, compose corporate emails, or whatever.) –jacobolus (t) 18:48, 16 July 2025 (UTC)

If your example is so complicated that readers can't obviously figure it out, please just explain with prose instead of letting them infer from tiny subtle differences of notation. As for non-mathematical typography: an ellipsis is ordinarily used for omitting words or sentences in the middle of running prose. In that context it doesn't really make too much sense to change their alignment. I imagine in other kinds of more structured two-dimensional notation other orientations or alignments of ellipses could be used (chemical equations?) but I don't know of any off hand. –jacobolus (t) 18:49, 16 July 2025 (UTC)

Please don't try to use ChatGPT to answer research questions. — I agree with you, despite that ChatGPT's answer to your statement suggests nuances and exceptions.

... explain with prose instead of letting them infer from tiny subtle differences of notation — generally I aim to do both. —Quantling (talk | contribs) 19:20, 16 July 2025 (UTC)

Is there a generating function

I noticed in the part where it said generating function it did not go in to terms on how to make Fn without writing the whole list, just in something else. Also , the connection to Pascal’s triangle is important and should be added 2A00:23C7:9910:EE01:295C:A9B5:540B:86B (talk) 09:39, 18 July 2025 (UTC)

I don't understand your question about the generating function. As for your other point, the article has a diagram and the line 'The Fibonacci numbers occur as the sums of binomial coefficients in the "shallow" diagonals of Pascal's triangle'. –jacobolus (t) 10:03, 18 July 2025 (UTC)

I have added the standard way for computing the generating series. D.Lazard (talk) 10:32, 18 July 2025 (UTC)

I edited the article before seeing this discussion. I apologize for that. Hopefully, the edit I made will be considered constructive nonetheless. If not, please revert or otherwise do the right thing. Thank you —Quantling (talk | contribs) 13:16, 18 July 2025 (UTC)

I removed the section since its understandable part contains only assertions that are elsewhere in the article. Flajolet et al. systematic use of generating series (even when they are never convergent) is interesting and very powerful, but I do not se any reason for mentioning it in this article. D.Lazard (talk) 21:13, 18 July 2025 (UTC)