Talk:Fibonacci sequence

Cite journal

@JayBeeEll: Feel free to ignore all this below if you've not already read it; I made a mistake. – Scyrme (talk) 19:35, 9 April 2023 (UTC)

Why revert a change to "cite journal"? It's a harmless change. Setting a specific type is helpful for bots and for error checking. (For instance, if the "journal" parameter is empty it will display an error for "cite journal" but not for "citation". In this case it's not empty; it's just an example.) In all likelihood the only reason it wasn't already used was because someone was adding references of different kinds and didn't want to figure out which references needed which template.

• Douady, S; Couder, Y (1996), "Phyllotaxis as a Dynamical Self Organizing Process" (PDF), Journal of Theoretical Biology, 178 (3): 255–74, doi:10.1006/jtbi.1996.0026, archived from the original (PDF) on 2006-05-26

• Douady, S; Couder, Y (1996), "Phyllotaxis as a Dynamical Self Organizing Process" (PDF), Journal of Theoretical Biology, 178 (3): 255–74, doi:10.1006/jtbi.1996.0026, archived from the original (PDF) on 2006-05-26

The first uses "cite journal", the second uses "citation". All the information is in the same order. – Scyrme (talk) 18:40, 9 April 2023 (UTC)

{{cite journal}} is Citation Style 1. {{citation}} is Citation Style 2. They differ. Most obviously in the. Way that. Citation Style 1. Breaks things up. With lots. Of. Periods. WP:CITEVAR is very clear that you should not be changing citation styles in this way without consensus. For those of us who use User:BrandonXLF/CitationStyleMarker.js to find inconsistent citation styles, your change is very annoying because it causes all of the citations to be flagged as inconsistent. Also your claim that this is helpful for bots and error checking seems dubious to me. The bots all know how to deal with both styles. —David Eppstein (talk) 18:45, 9 April 2023 (UTC)

The documentation indicates that this can be toggled with a parameter. Setting |mode=cs2 should make them identical. – Scyrme (talk) 18:50, 9 April 2023 (UTC)

@David Eppstein: Regarding bots, I've seen some change the type (from "web" to "journal" or whatever, not CS1 to CS2), so I assumed that it would reduce the task queue if nothing else. I wasn't aware that {{citation}} used a different style by default to the other citation templates. Surprises me that it does, to be honest. That seems needlessly confusing.

Does setting the "mode" parameter still result in that annoyance? – Scyrme (talk) 18:57, 9 April 2023 (UTC)

Not that specific annoyance. But if you keep going around making cosmetic changes to the source code of articles without doing anything actually constructive, it will still continue to annoy me by cluttering my watchlist and wasting my time checking the edits. There's also an argument that it's still in violation of CITEVAR, because CITEVAR controls the appearance of citation sourcing (for instance, whether the references are list-defined or inline), not just the readers' view of citations. The article currently uses one template consistently for its citations, and you would be going against that consistency. Changing cite web to cite journal is different: that's staying within the same family of templates and merely (one hopes) making the type more accurate. —David Eppstein (talk) 19:03, 9 April 2023 (UTC)

Like I said, I wasn't aware that {{citation}} wasn't in the same family of templates. I thought I was just making the template more accurate. Now that I know, I won't mix them up. – Scyrme (talk) 19:24, 9 April 2023 (UTC)

Every k-th Fibonacci number

Does anyone have a reference to (or knowledge of) linear recurrence formulas for every-k-th term of the Fibonacci sequence? For example with k = 2, we can compute every other value recursively with F_n+4 = 3F_n+2 − F_n, for any integer n ≥ 0. Likewise with k = 3, we can compute every third value F_n+6 = 4F_n+3 + F_n. With k = 4, we can compute every fourth value with F_n+8 = 7F_n+4 − F_n. —Quantling (talk | contribs) 15:01, 10 May 2023 (UTC)

Apply the roots of unity filter to the GF to get the GF for the multisection. The denominator will be $\prod _{i=0}^{k-1}(1-\omega ^{k}x-\omega ^{2k}x^{2})$ where $\omega$ is a primitive complex k-th root of unity, and its coefficients are the coefficients of the recurrence (by general theory of when ordinary GFs are rational functions). --JBL (talk) 17:55, 10 May 2023 (UTC)

It seems to be the case that

$${\begin{aligned}F_{n+2k}&=\alpha F_{n+k}+\beta F_{n}\mathrm {,with} \\\alpha &={\frac {F_{2k}}{F_{k}}}\\\beta &=F_{2k+1}-{\frac {F_{2k}}{F_{k}}}F_{k+1}\,.\end{aligned}}$$

But, even if this qualifies as simple calculation, we are going to need proof of notability. —Quantling (talk | contribs) 17:55, 10 May 2023 (UTC)

Those coefficients are a Lucas number and $\pm 1$. Personally I would not be inclined to include this for reasons of due weight. --JBL (talk) 18:00, 10 May 2023 (UTC)

Thank you. So that makes it

One can compute every k-th Fibonacci number with the recurrence F_n+2k = L_k F_n+k − (−1)^k F_n, where n is any non-negative integer and L_k = F_k+1 + F_k−1 is a Lucas number.

That simplifies the presentation. (Still need indication that it has due weight.) —Quantling (talk | contribs) 18:29, 10 May 2023 (UTC)

I am going ahead with a WP:BRD edit, with the hope that it draws additional editors to this discussion. —Quantling (talk | contribs) 18:44, 11 May 2023 (UTC)

You added it to the article without any source at all. Please do not do that. It needs a published source. —David Eppstein (talk) 20:18, 11 May 2023 (UTC)

If someone has a way to search back issues of Fibonacci Quarterly, there's absolutely no way this hasn't been published before. I'm not at all worried about this being unverifiable. But even if sources are hunted down, I still wonder whether/why this identity is noteworthy among the dozens of similar identities that could be written down. –jacobolus (t) 05:14, 12 May 2023 (UTC)

For example, it's exercise 52–53 in https://doi.org/10.1002/9781118033067.ch5 and theorems 5, 7 in https://doi.org/10.2307/3619903 –jacobolus (t) 09:51, 12 May 2023 (UTC)

If we can find discussion in Fibonacci Quarterly, perhaps that will signal noteworthiness as well. Unfortunately, I do not have access. —Quantling (talk | contribs) 13:28, 12 May 2023 (UTC)

You see the Math. Gazette article right?

Koshy, Thomas (1998). "82.55 New Fibonacci and Lucas Identities". Mathematical Gazette. 82 (495): 481–484. doi:10.2307/3619903. JSTOR 3619903.

It has Theorems 4–7:

${\begin{aligned}L_{2m+n}-(-1)^{m}L_{n}&=5F_{m}F_{m+n},\\[3mu]F_{2m+n}-(-1)^{m}F_{n}&=F_{m}L_{m+n},\\[3mu]L_{2m+n}+(-1)^{m}L_{n}&=L_{m}L_{m+n},\\[3mu]F_{2m+n}+(-1)^{m}F_{n}&=L_{m}F_{m+n}.\end{aligned}}$

But the main question is: why do you think this identity is important out of the dozens of similar identities one could write down? –jacobolus (t) 15:23, 12 May 2023 (UTC)

Why does it appeal to me?: because I find it interesting that a sequence that is every k-th term of the Fibonacci sequence obeys a linear recurrence (and that the linear recurrence has easily computed coefficients). Of the above, only Theorem 7 is about such a sequence. While I find pretty much all math to be at least somewhat interesting, I don't (yet) see as much beauty in the other three theorems. FWIW, there are formulas that are already in the article that I also find less interesting than this subsequence formula.

But obviously this isn't about me, and unfortunately I don't have access to pretty much anything proprietary online. If there is an established author who shows interest similar to mine then I am hopeful that we could build consensus around including this formula. —Quantling (talk | contribs) 15:49, 12 May 2023 (UTC)

Do you know about the "Wikipedia Library"? –jacobolus (t) 18:50, 12 May 2023 (UTC)

No. Nice. —Quantling (talk | contribs) 19:49, 12 May 2023 (UTC)

It looks both @JBL and @jacobolus are against including this text for reasons of due weight. Absent new evidence, it looks like the proposed edit is not going happen. FWIW, maybe @David Eppstein's concern for a reference has been satisfied. Thank you for the discussion —Quantling (talk | contribs) 13:56, 16 May 2023 (UTC)

I am ambivalent about it. I haven't thought much at all about this particular article. But in general, it's hard to figure out which identities are interesting enough to tell readers about. –jacobolus (t) 14:52, 16 May 2023 (UTC)

One other highly cited paper you may want to take a look at is Rabinowitz (1996) doi:10.1007/978-94-009-0223-7_33 http://stanleyrabinowitz.com/download/algorithmicfib.pdf –jacobolus (t) 19:35, 16 May 2023 (UTC)

Fibonacci Approximation Function

A function that approximates the Fibonacci sequence is the following: y = (0.003)x^6 - (0.0064)x^5 + (0.0701)x^4 - (0.3435)x^3 + (1.0052)x^2 -(0.6578)x + 0.9895

Try and plot it on Octave or in any other software.

~_Roberto::Barone_~ 95.234.175.68 (talk) 15:14, 18 September 2023 (UTC)

It is possible to come up with many polynomials that accurately approximate a few of the Fibonacci numbers. (See polynomial fitting, generally.) However, regardless of the polynomial p(x), it is the case that

$$\lim _{x\rightarrow +\infty }{\frac {p(x)}{\phi ^{x}}}=0\,,$$

and thus the polynomial will fall woefully short of the Fibonacci sequence for large enough values of x. —Quantling (talk | contribs) 15:38, 18 September 2023 (UTC)

Thanks for your reply.

I know... Mine was just an approximate attempt for the sequence.

~_Roberto::Barone_~ 95.234.175.68 (talk) 16:01, 18 September 2023 (UTC)

Pictures in article

The first two pictures in the article illustrate some features of the squares of Fibonacci numbers. Don't we have any pictures to illustrate the actual Fibonacci numbers instead of their squares? Eddi (Talk) 16:32, 2 October 2023 (UTC)

With the addition of some thickened / darkened lines, the first picture could emphasize non-squares. The lines that I would thicken are

• the topmost line, of length 34,
• the rightmost line, of length 21,
• the line from the lower right corner, going left for a length of 13,
• from there going up for a length of 8,
• from there going right for a length of 5,
• then down 3,
• then left 2,
• then up 1, and
• then right 1.

—Quantling (talk | contribs) 20:41, 2 October 2023 (UTC)

The areas of the squares in the figure are squares of Fibonacci numbers, but the properties illustrated are not particularly areal. --JBL (talk) 17:05, 3 October 2023 (UTC)

It's a fair point that these images show more or less the same thing twice, and one of the two could be substituted with another type of image without losing much. –jacobolus (t) 17:13, 3 October 2023 (UTC)

Sure, I agree. --JBL (talk) 20:33, 3 October 2023 (UTC)

Intro paragraph

There is not only one Fibonacci sequence. The sequence 2,5,7,12,19,... is a Fibonacci sequence. So I do not see why the introduction should refer to "the" Fibonacci sequence. Shouldn't it describe "a" Fibonacci sequence?

Also, it is not true that every number of a Fibonacci sequence is the sum of the two predecessors (as explained in the introduction paragraph) since the first two numbers in a sequence do not have two predecessors. Shouldn't the language be made precise?

I attempted to correct these two issues back in October 2023, but Jaybee didn't like my edits and reverted them, saying that I should not have done so. Why? Majfoster (talk) 06:37, 7 December 2023 (UTC)

First, because Wikipedia articles must be based on the consensus of mainstream published sources, not on the idiosyncratic views of individual editors. Second, because this article is about the usual Fibonacci sequence, not other sequences defined from the same recurrence. We have a separate article for that: Generalizations of Fibonacci numbers. —David Eppstein (talk) 07:27, 7 December 2023 (UTC)

Thank you for the feedback. I see now there is another page for the generalized sequence. I should have done my due diligence to see if it existed before making a fuss. Majfoster (talk) 16:44, 7 December 2023 (UTC)

I agree that starting with other than 0, 1 should be in the other article.

I've boldly made an edit to reflect the other point you make; the lead sentence now reads "In mathematics, the Fibonacci sequence is a sequence starting with 0 and 1 in which each subsequent number is the sum of the two preceding ones." For the very first sentence that may be too much information, but let's see what other editors think. —Quantling (talk | contribs) 15:08, 7 December 2023 (UTC)

Excellent, I am am satisfied! Majfoster (talk) 16:45, 7 December 2023 (UTC)

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 —Quantling (talk | contribs) 14:39, 2 February 2024 (UTC)

No big deal indeed, so no problem with me. Afaiac, go ahead . - 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. D.Lazard (talk) 16:30, 2 February 2024 (UTC)

I agree. Of course we can only make changes where it makes sense. - DVdm (talk) 16:41, 2 February 2024 (UTC)

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 $a+b\varphi$ have "individual Fibonacci numbers" as their coefficients, as described in Fibonacci sequence § Decomposition of powers. –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. D.Lazard (talk) 17:55, 2 February 2024 (UTC)