Talk:Fibonacci sequence/Archive 2

Archive 1

Archive 2

Archive 3

Archive 4

Gopala / Hemachandra

I've removed the clause from the introduction that says that the numbers are also called the "Gopala-Hemachandra numbers". The page already mentions that Fibonacci was anticipated by Gopala and Hemachandra, and I find no evidence that the numbers are actually called the "Gopala-Hemachandra numbers".

I'm also going to redirect the Gopala-Hemachandra numbers article to this one, since the two phrases mean the same thing and that article contains nothing that isn't already in this one.

-- Dominus 14:16, 11 Nov 2004 (UTC)

Addendum: even the external research paper linked to from the Gopala-Hemachandra numbers page does not refer to the numbers as the "Gopala-Hemachandra numbers". It says "The numbers in the sequence are called Fibonacci numbers." The phrase "Gopala-Hemachandra numbers" does not appear in that paper.

-- Dominus 14:18, 11 Nov 2004 (UTC)

i agree that the internal reaseach paper linked to from the page definatley does not refer to the numbers as gopala-hemachandra numbers.i have checked twice over and it does not apper in the paper. —Preceding unsigned comment added by 213.1.35.46 (talk) 14:35, 25 September 2007 (UTC)

I note we still have a page Gopala-Hemachandra number (no s at the end) which is not a redirect. I've now redirected it to here. --Salix alba (talk) 14:53, 25 September 2007 (UTC)

Bartok

From the "Application" section: "It is commonly thought that the first movement of Béla Bartók's Music for Strings, Percussion, and Celesta was structured using Fibonacci numbers."

Well maybe it is commonly thought, but that doesn't mean it is true. Until someone can come up with an explanation on why that movement has 88 bars and not 89 as the Fibonacci sequence would suggest, I would like to see this part removed from the article. NguyenVanThoc 22:41, 30 November 2007 (UTC)

Formula

While these nifty bignum formulas are nice and all, I think it would be very nice to have the actual formula for calculating them. The math isn't that hard to do by hand, because of cancelling pieces. I think that the article needs it because it is the non recursive form of it.

See the Closed Form Expression, which translates to:

$F\left(n\right)=({1 \over {\sqrt {5}}})(({1+{\sqrt {5}} \over 2})^{n}-({1-{\sqrt {5}} \over 2})^{n}))$

Courtesy of Posamentier and Lehmann -Dagordon01 16:40, 1 December 2007 (UTC)

Formula of finding fibonacci number

lets say a = sqrt(5), then: F(n) = ((a+1)^(n+1)-(a-1))/(2^(n+1)*a)

See the discussion above about "Formula" and the reference to Posamentier and Lehmann -Dagordon01 16:52, 1 December 2007 (UTC)

Popular Culture

This section is VERY vague. Should some examples be given? —Preceding unsigned comment added by BrettxPW (talk • contribs) 20:52, 10 December 2007 (UTC)

Identity for doubling n

I added the actual identity for doubling n. I think the formula for F_{2n+k) needs a reference or something since I have never seen that before. The reference provided right below that does NOT contain that identity and indeed contains an identity that is completely wrong: F_2n=F_{n}^2+F_{n-1}^2. I believe that reference should be removed. I also don't see how it reduces to the F_2n formula when k = 0. Also, it should definitely not say for all integers k and n because it doesn't make sense if n<0 or k<3. (SlaterDeterminant (talk) 16:44, 3 January 2008 (UTC))

I fixed the formula that you added for F_2n. The F_2n+k formula looks fine to me - it is just a special case of Formula 47 from the MathWorld page. When k=0 you have F_k=0, F_k-1=1 and F_k-2=-1, so you get F_2n = 2F_n+1F_n - F_n² as expected. Gandalf61 (talk) 17:17, 3 January 2008 (UTC)

Proof by induction

Why haven’t you completed your Proof by induction of Binet’s formula? You’ve shown its true for 0 and 1.I think you now need to show that if it’s true for n and n+1 then it is also true for n+2,the dominoes topple, and you’ve proved it for all the natural numbers. I’ve just tried to do this on a bit of paper and I can’t.It certainly isn’t so obvious you can just leave it out! —Preceding unsigned comment added by 91.107.165.60 (talk) 21:23, 9 January 2008 (UTC)

Duh! I’ve just seen how to do it. It is pretty obvious but someone who can write Latex ought to put it up. —Preceding unsigned comment added by 91.107.165.60 (talk) 21:39, 9 January 2008 (UTC)

I’ve just tried to do it by cutting and pasting Lyx code but that doesn’t work. I get “parsing error”. I’m not going to learn all the bloody code, someone else will have to do it. Anyone who thinks for 5 minutes will see how the proof works anyway. It just annoys me it isn’t completed. Dave59 (talk) 23:01, 9 January 2008 (UTC)

let P(n) be the variable proposition $F_{n}={\frac {\phi ^{n}-\psi ^{n}}{\sqrt {5}}}$

P(n+1) is $F_{n+1}={\frac {\phi ^{n+1}-\psi ^{n+1}}{\sqrt {5}}}$

P(n+2) is $F_{n+2}={\frac {\phi ^{n+2}-\psi ^{n+2}}{\sqrt {5}}}$

Now $F_{n+2}=F_{n+1}+F_{n}$

Therefore $F_{n+2}={\frac {\phi ^{n}-\psi ^{n}}{\sqrt {5}}}+{\frac {\phi ^{n+1}-\psi ^{n+1}}{\sqrt {5}}}={\frac {\left(\phi ^{n}+\phi ^{n+1}\right)-\left(\psi ^{n}+\psi ^{n+1}\right)}{\sqrt {5}}}$

we have allready shown $\phi ^{n+2}=\phi ^{n+1}+\phi ^{n}\qquad and\qquad \psi ^{n+2}=\psi ^{n+1}+\psi ^{n}$

Therefore $F_{n+2}={\frac {\phi ^{n+2}-\psi ^{n+2}}{\sqrt {5}}}$

So $P(n)\qquad and\qquad P(n+1)\Rightarrow P(n+2)$

we have already shown P(0) and P(1) are true

Therfore by mathematical induction the proposition is proved for all natural numbers.(or for all the natural numbers plus zero if you want to be really pedantic)

Code a damn site harder than the maths

Dave. —Preceding unsigned comment added by 91.105.18.197 (talk) 12:42, 11 January 2008 (UTC)

This seems already explained in Fibonacci_number#Proof_by_induction.--Patrick (talk) 13:46, 11 January 2008 (UTC)

It probably says enough for a mathematician to understand the drift straight away. However it is not a formal proof by induction and I didn’t understand it the first time I read it. This just dots the i’s and crosses the t’s. This is pure maths and I feel we ought to be precise. I have used slightly different notation to the main article. I’m unfamiliar with Latex and this took me ages to do. I’m not even going to try to integrate it into the main article. It is probably true that most of the people who are going to read the article don’t need it but it might be useful for people who are just learning proof by induction and want to see a few examples. Dave59 (talk) 15:37, 11 January 2008 (UTC)

Fibonacci sequence

This article should be called Fibonacci sequence and not Fibonacci number. A Fibonacci number is meaningless out of the context of its sequence. If I asked you "what is 21?", nobody would say "the Fibonacci number after 13". But if I asked "what is 1, 1, 2, 3, 5, 8, 13, 21...?, I'd have a much greater chance of hearing "Fibonacci sequence". This article should be moved to Fibonacci sequence over the redirect, and Fibonacci number should redirect to Fibonacci sequence. TableManners^C·U·T 06:05, 18 January 2008 (UTC)

I was going to say it's commonly called "numbers" by everyone in the world, but then I looked at the interwiki links: bg, cs, eo, pt, ru - Numbers. ca, de, el, es, fr, it, scn, sk, tr, uk - Sequence. Still, I've mostly seen it as "numbers" in English - for example that's how it's called on the Integer Sequences site [1] and, for another example, Wolfram's Mathworld defines the Sequence [2] as "see Fibonacci Number". The Marriam-Webster dictionary of the English Language has the entry for numbers [3] but not sequence, while American Heritage Dictionary has both and essentially says "See Sequence" for Number: [4] and [5]. Doesn't look like there is an agreement. --Cubbi (talk) 12:23, 18 January 2008 (UTC)

Though I'm not sure it's supported by Cubbi's post, isn't "sequence" a alightly technical mathematician's way of putting it, and "numbers" what the man in the street would say? Fibonacci numbers are rather insignificant in professional math, but play a quite significant role in popular math, recreational math. I'm for keeping the article at "numbers".--Niels Ø (noe) (talk) 13:19, 18 January 2008 (UTC)

Conditions such as "if n is a Fibonacci number" naturally arise, independent of any overt connexion to the sequence, often enough that I disagree with TableManners: they are a meaningful set or class of numbers. —Tamfang (talk) 23:07, 22 January 2008 (UTC)

• Just passing by, figured I'd share my 2 cents. Google results:

"Fibonacci sequence" (with quotes): ~186,000

"Fibonacci number" (with quotes): ~86,000

"Fibonacci numbers" (with quotes): ~216,000

Therefore, I propose a move to Fibonacci numbers with Fibonacci sequence and Fibonacci number redirecting to that title. FireCrotch (talk) 15:18, 28 January 2008 (UTC)

Current convention for Wikipedia articles on integer sequences is to name them xxx number or xxx sequence but never xxx numbers - see Category:Integer sequences for many examples. This follows Wikipedia:Naming conventions, which says "In general only create page titles that are in the singular, unless that noun is always in a plural form in English (such as scissors or trousers)". Gandalf61 (talk) 15:32, 28 January 2008 (UTC)

Thank you for pointing that out, Gandolf61. In that case, I suggest that it be renamed to "Fibonacci sequence". Now that I think about it, of course "Fibonacci numbers" is going to have more results - it includes all pages that contain "Fibonacci number" as well! FireCrotch (talk) 04:00, 29 January 2008 (UTC)

Why two separate "Pythagorean triple" sections?

"Pythagorean triples of Fibonacci numbers" is the subject of two separate sections of this article:

• "Right triangles," and
• "Pythagorean triples"

I'm not sure what might be the most parsimonious/harmonious way to do it, but wouldn't it be best to somehow merge these sections?
—Wikiscient— 11:01, 12 March 2008 (UTC)

I merged them. —David Eppstein (talk) 14:46, 12 March 2008 (UTC)

Bees - Got it backwards I think

Males only come from mated bees (how can a female introduce a male chromosome?) The logic in how it relates to the Fibonacci sequence is still the same, but I think male and female were switched in the logic. I have changed it, and if you find I am wrong (with references of course) feel free to undo my switch. I only found this link as a reference for now, maybe will come back later with more. --Billy Nair (talk) 16:39, 16 March 2008 (UTC)

I am pretty sure the same is with chickens, unfertalized eggs will be female, and only fertalized eggs have the chance to be male, I don't know if it is always male, but need a male to get a male. --Billy Nair (talk) 16:41, 16 March 2008 (UTC)

Sorry, but the article was right - male bees, also known as drones, develop from unfertilised eggs. See our drone (bee) article, which explains how the genetics works. I have changed the article back to how it was. Gandalf61 (talk) 17:35, 16 March 2008 (UTC)

Birds, I believe, have a system analogous to the XY of mammals but the other way around: a bird with matching chromosomes is male, one with differing sex chromosomes is female; but a haploid (unfertilized) egg won't develop at all. —Tamfang (talk) 22:22, 3 April 2008 (UTC)

Labels in the list of values

I just reverted a change by Virginia-American (talk · contribs) that replaced the list of values near the start of the article,

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, ...

by a list in which each value is labeled,

F₀ = 0, F₁ = 1, F₂ = 1, F₃ = 2, F₄ = 3, F₅ = 5,

F₆ = 8, F₇ = 13, F₈ = 21, F₉ = 34, F₁₀ = 55,

F₁₁ = 89, F₁₂ = 144, F₁₃ = 233, F₁₄ = 377, F₁₅ = 610,

F₁₆ = 987, F₁₇ = 1597, F₁₈ = 2584, F₁₉ = 4181, F₂₀ = 6765,

F₂₁ = 10946, F₂₂ = 17711, F₂₃ = 28657, F₂₄ = 46368, F₂₅ = 75025,

F₂₆ = 121393, ...

I think the labels make the list completely unreadable. But since this is a content disagreement rather than something more clear-cut, I thought I'd bring it here for further discussion, if there is any. —David Eppstein (talk) 15:40, 22 March 2008 (UTC)

Obviously I disagree. I got annoyed trying to verify some of the formulas and having to count to see what value of n corresponded to which fibonacci number. Actually, a table of some sort is probably the best way to display the first few values. Virginia-American (talk) 15:50, 22 March 2008 (UTC)

I made a table. I agree that the n is needed.--Patrick (talk) 16:13, 22 March 2008 (UTC)

Sorry, but I reverted the table. If you have TOC switched on the table runs down the left hand side of the TOC and looks just awful. And it makes the TOC appear in the middle of the lead section, for some reason. I wouldn't object to a table if (a) it is multi-column so it takes up less vertical space and (b) it can be arranged so as not to overlap the TOC. Gandalf61 (talk) 18:15, 22 March 2008 (UTC)

I made a table going horizontally instead of vertically. I think it looks better, but in order to work with possibly narrow browser windows I truncated the sequence earlier (21 terms). —David Eppstein (talk) 18:59, 22 March 2008 (UTC)

Yes, that looks better. Good job. Gandalf61 (talk) 21:16, 22 March 2008 (UTC)

Thanks for the table - great reference-ability on that. Bugtank (talk) 04:47, 6 April 2008 (UTC)

The most notable property of the Fibonacci sequence is the ratio converting to $\varphi$ = 1.6180339887... Would it be worth the space to put something like the below table in a section (not the lead)? There have been complaints that the article is complicated. The table is simple and could come before more complicated parts. PrimeHunter (talk) 21:51, 22 March 2008 (UTC)

Fibonacci numbers

n	Fn	Factorization	Fn / Fn-1	abs(Fn / Fn-1 − $\varphi$)
0	0
1	1	1
2	1	1	1	0.6180339887
3	2	2	2	0.3819660113
4	3	3	1.5	0.1180339887
5	5	5	1.6666666667	0.0486326779
6	8	23	1.6	0.0180339887
7	13	13	1.625	0.0069660112
8	21	3·7	1.6153846154	0.0026493733
9	34	2·17	1.6190476190	0.0010136302
10	55	5·11	1.6176470588	0.0003869299
11	89	89	1.6181818182	0.0001478294
12	144	24·32	1.6179775281	0.0000564606
13	233	233	1.6180555556	0.0000215668
14	377	13·29	1.6180257511	0.0000082376
15	610	2·5·61	1.6180371353	0.0000031465
16	987	3·7·47	1.6180327869	0.0000012018
17	1597	1597	1.6180344478	0.0000004590
18	2584	23·17·19	1.6180338134	0.0000001753
19	4181	37·113	1.6180340557	0.0000000669
20	6765	3·5·11·41	1.6180339632	0.0000000255
21	10946	2·13·421	1.6180339985	0.0000000097
22	17711	89·199	1.6180339850	0.0000000037
23	28657	28657	1.6180339902	0.0000000014
24	46368	25·32·7·23	1.6180339882	0.0000000005
25	75025	52·3001	1.6180339890	0.0000000002

PrimeHunter said "The most notable property of the Fibonacci sequence is the ratio converting to $\varphi$ = 1.6180339887... ".

Not really. It doesn't matter what the starting values are, as long as a sequence has the same recurrence F_n+1 = F_n + F_n-1 as the Fibonacci sequence, the ratios converge to phi. I think the fact that the convergents of the continued fraction for phi are the ratios of consecutive F_n s is more notable. Virginia-American (talk) 17:35, 27 March 2008 (UTC)

I meant notable as in Wikipedia:Notability, meaning there are lots of sources about it. I think your property is mentioned relatively rarely. PrimeHunter (talk) 17:42, 27 March 2008 (UTC)

Error in proof

I fixed an error in the proof of the third identity. Paul August ☎ 05:44, 9 April 2008 (UTC)