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== 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''. <
:: 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 [http://www.research.att.com/~njas/sequences/A000045] and, for another example, Wolfram's Mathworld defines the Sequence [http://mathworld.wolfram.com/FibonacciSequence.html] as "see Fibonacci Number". The Marriam-Webster dictionary of the English Language has the entry for numbers [http://www.m-w.com/dictionary/Fibonacci%20number] but not sequence, while American Heritage Dictionary has both and essentially says "See Sequence" for Number: [http://www.bartleby.com/61/0/F0100000.html] and [http://www.bartleby.com/61/1/F0100100.html]. Doesn't look like there is an agreement. --[[User:Cubbi|Cubbi]] ([[User talk: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".--[[User:Noe|Niels Ø (noe)]] ([[User talk:Noe|talk]]) 13:19, 18 January 2008 (UTC)
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This is an example of how to print a Fibonacci series in [[C (programming language)|C]]:
<
#include<stdio.h>
#include<conio.h>
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return fib(n-1)+fib(n-2);
}
</syntaxhighlight>
== MathWorld Fibonacci article should be linked ==
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:The quoted caption at [[Fibonacci number#Relation to the golden ratio]] looks right to me. 1 / (1/φ) = φ (this holds for any non-zero number and not just φ). And more generally, (1/φ<sup>''n''</sup>) / (1/φ<sup>''n+1''</sup>) = φ (also holds for all non-zero numbers). In addition, φ has the special property 1/φ = φ−1. [[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 01:15, 2 May 2011 (UTC)
== "Gopala-Hemachandra Numbers" ==
My remarks on [[Talk:Gopala–Hemachandra number]] seem to be relevant, so I am copying them here. I said:
:There does not seem to be any source for the main claim of this article, namely that "A Gopala–Hemachandra number is a term in a sequence of the form …." The sources cited in the article do ''not'' state this. I cannot find any indication that anyone actually uses the term "Gopala–Hemachandra number" in this way, or indeed for anything else.
:I am not disputing that Gopala and Hemchandra dicussed the Fibonacci series before Fibonacci did; I agree that that is well-established. My only complaint is with the claim that the term "Gopala–Hemachandra number" is a recognized term.
:Two of the cited sources refer to "Gopala–Hemachandra codes". I don't think these two sources are enough to establish that the term is widely used. One of the two sources cited for this is self-published, and does not meet Wikipedia's standards for [[WP:RELIABLE|reliable sources]]. —[[User:Dominus|Mark Dominus]] ([[User talk:Dominus|talk]]) 15:13, 28 March 2011 (UTC)
The J.H. Thomas source does use "Gopala-Hemachandra sequence" to refer to a general sequence that obeys the recursion {{mvar|a{{su|b={{mvar|n}}}}}} = {{mvar|a{{su|b={{mvar|n}}-2}}}} + {{mvar|a{{su|b={{mvar|n}}-1}}}}, but it is self-published. I have not seen the Basu-Prasad paper yet; I have written to Professor Basu asking for a copy, but she has not yet replied. But even if it does use "Gopala-Hemachandra sequence" or "Gopala-Hemachandra number", it's only a single paper; the term is clearly not in widespread use, and I think it is inappropriate to add it to this article as if it were a common phrase. —[[User:Dominus|Mark Dominus]] ([[User talk:Dominus|talk]]) 16:00, 29 March 2011 (UTC)
:Just about everyone in the world says "Fibonacci number". I hate it when people claim we can't use the standard terminology for some newly discovered historical reason. The result, when the new name is used, is that finding literature becomes substantially and unnecessarily more difficult. Attempts at changing names happen regularly, usually when a different European mathematician is claimed to precede the one whose name is used. Often, the justification for the claim is rather unclear, and usually it does not change existing terminology. Thank goodness. Changing very well-established terminology ought to be done only for very good reasons. [[User:Zaslav|Zaslav]] ([[User talk:Zaslav|talk]]) 06:26, 28 June 2011 (UTC)
::Anyway the standard here is to name things what they are named, not what they should be named. See also [[Stigler's law of eponymy]]. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 06:43, 28 June 2011 (UTC)
== How a bit of information on the complex plane? ==
If you use the equations on this page for i, what is the i<sup>th</sup> Fibonacci number? [[User:Robo37|Robo37]] ([[User talk:Robo37|talk]]) 09:43, 6 July 2011 (UTC)
:I think it's 0.221247712 + 0.299699204i. Do you think this should be included in the artical?
::Can you show the details of your calculation ? [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 13:09, 8 July 2011 (UTC)
:::The details of my calculation?... I entered (((1+sqrt 5)/2)^i-(-(1+sqrt 5)/2))^i)/sqrt 5 into Google calculator... I am no mathematician, sorry, but maybe if you can look into how Google calculator got to this you might be able to change that into some king of closed solution involving e and/or pi. [[User:Robo37|Robo37]] ([[User talk:Robo37|talk]]) 13:38, 11 July 2011 (UTC)
::::Oops, wrong formular. The i<sup>th</sup> Fibonacci number is ''0.379294534 + 0.215939518 i'', which I got by ((1+sqrt 5)/2)^i-((-1)^i/((1+sqrt 5)/2)^i)))/sqrt 5.
== Fibonacci root? ==
What is x when F (x) = n? [[User:Robo37|Robo37]] ([[User talk:Robo37|talk]]) 09:57, 14 July 2011 (UTC)
:See [[Fibonacci number#Recognizing Fibonacci numbers]]. [[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 11:45, 14 July 2011 (UTC)
::Thanks, is it at all expressable without using uncommon logs? [[User:Robo37|Robo37]] ([[User talk:Robo37|talk]]) 17:58, 15 July 2011 (UTC)
:::You can always change a logarithm base to any other with the formula at [[Logarithm#Change of base]]. [[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 02:08, 16 July 2011 (UTC)
::::Wait, how come when I enter log ((21*sqrt 5)+0.5)/log ((1+sqrt 5)/2) into Google calculator I get 8.02106857? [[User:Robo37|Robo37]] ([[User talk:Robo37|talk]]) 07:42, 16 July 2011 (UTC)
:::::Because that is the (approximate) value of that expression. Note that to find the Fibonacci index you have to take the [[Floor and ceiling functions|floor]] (nearest integer less than or equal to) this value, which gives you 8. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 14:56, 16 July 2011 (UTC)
::::::The difference to the real value converges to 0. In [[PARI/GP]] for powers of two:
<pre>
? for(n=1,8,print(2^n": "log((fibonacci(2^n)*sqrt(5))+0.5)/log((1+sqrt(5))/2)))
2: 2.09163988209225188766928942555474042584550500130492803616758
4: 4.10467845877076686103432829279967108337857360119089528434855
8: 8.02106857224470714263573083533459921808575680298676770865263
16: 16.0004703147334956159652160508455984425808456893171492670951
32: 32.0000002133187473657459828226685085556081114874256292899634
64: 64.0000000000000437950208463788551755832830904781992970335194
128: 128.000000000000000000000000001845932267159675589494339813885
256: 256.000000000000000000000000000000000000000000000000000003279
</pre>
::::::[[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 16:25, 16 July 2011 (UTC)
:::::::Oh, right, thanks. What is the (closed) exact formular, then? [[User:Robo37|Robo37]] ([[User talk:Robo37|talk]]) 17:08, 16 July 2011 (UTC)
::::::::Well, if you want an expression that does not involve the floor function then it is
:::::::::<math>n = \log_\varphi\bigg(\frac{\sqrt{5F_n^2\pm4}+\sqrt{5}F_n}{2}\bigg)</math>
::::::::where you choose the sign (plus or minus) that makes <math>\sqrt{5F_n^2\pm4}</math> an integer. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 19:47, 16 July 2011 (UTC)
:::::::::Thanks, but that get's to (log (((5*(((1+sqrt 5)/2)^21-((-1)^21/((1+sqrt 5)/2)^21)))/sqrt 5)))+4)+((sqrt 5)*((1+sqrt 5)/2)^21-((-1)^21/((1+sqrt 5)/2)^21)))/sqrt 5)))/2))/(log ((1+sqrt 5)/2)) and when I enter it into Goodle calculater I get 58567.8366. With an equation as massive as that I'm bound to make an error somewhere along the line but there must be a shorter expression than that I can use '''without''' using the Fibonacci function itself. [[User:Robo37|Robo37]] ([[User talk:Robo37|talk]]) 20:17, 16 July 2011 (UTC)
::::::::::F<sub>8</sub> = 21, so
:::::::::::<math>\log_\varphi\bigg(\frac{\sqrt{5 \times 21^2+4}+21\sqrt{5} }{2}\bigg)
=\log_\varphi\bigg(\frac{\sqrt{2209}+21\sqrt{5}}{2}\bigg)=\log_\varphi\bigg(\frac{47+21\sqrt{5}}{2}\bigg)=8</math>
::::::::::Without using the floor function, that's as simple as it gets. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 21:33, 16 July 2011 (UTC)
:::::::::::Here is a valid ASCII expression to compute x when F (x) = n:
:::::::::::x = log((sqrt(5*n^2+4)+sqrt(5)*n)/2)/log(((1+sqrt(5))/2))
:::::::::::Replace +4 by -4 if required to get an integer (this happens when x is odd). [[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 23:02, 16 July 2011 (UTC)
::::::::::::Thanks for that, we got there eventually. So, as I don't see <math>\log_\varphi\bigg(\frac{\sqrt{5 \times n^2+4}+n\sqrt{5} }{2}\bigg)=x</math> and <math>\log_\varphi\bigg(\frac{\sqrt{5 \times n^2-4}+n\sqrt{5} }{2}\bigg)=x</math> anywhere on the artical, maybe it should be included somewhere? Anyway, thanks again, I appreciate the help. [[User:Robo37|Robo37]] ([[User talk:Robo37|talk]]) 09:06, 17 July 2011 (UTC)
:::::::::::::Sorry about this, but is there a universal function that for that that isn't dependant on whether x is odd or even? [[User:Robo37|Robo37]] ([[User talk:Robo37|talk]]) 22:38, 29 July 2011 (UTC)
== On entering decimals into the Fibonacci function.. ==
We have F(x)=((1+sqrt 5)/2)^(x)-((-1)^(x)/((1+sqrt 5)/2)^(x))))/sqrt 5, which indeed satifies any integar as:
* ((1+sqrt 5)/2)^(0)-((-1)^(0)/((1+sqrt 5)/2)^(0))))/sqrt 5=0
* ((1+sqrt 5)/2)^(1)-((-1)^(1)/((1+sqrt 5)/2)^(1))))/sqrt 5=1
* ((1+sqrt 5)/2)^(2)-((-1)^(2)/((1+sqrt 5)/2)^(2))))/sqrt 5=1
* ((1+sqrt 5)/2)^(3)-((-1)^(3)/((1+sqrt 5)/2)^(3))))/sqrt 5=2
* ((1+sqrt 5)/2)^(4)-((-1)^(4)/((1+sqrt 5)/2)^(4))))/sqrt 5=3
* ((1+sqrt 5)/2)^(5)-((-1)^(5)/((1+sqrt 5)/2)^(5))))/sqrt 5=5
* ((1+sqrt 5)/2)^(6)-((-1)^(6)/((1+sqrt 5)/2)^(6))))/sqrt 5=8...
...and so on, so it thereby would make sense to use the same forumular for decimals.
Using Google calculator, we have:
{| class="wikitable"
|-
! x !! ((1+sqrt 5)/2)^(1/x)-((-1)^(1/x)/((1+sqrt 5)/2)^(1/x))))/sqrt 5
|-
| 1 || 1
|-
| 2 || 0.568864481 - 0.351577584 i
|-
| 3 || 0.905958432
|-
| 4 || 0.224000793 - 0.280383911 i
|-
| 5 || 0.898572747
|-
| 6 || 0.127109339 - 0.206373405 i
|-
| 7 || 0.896541471
|-
| 8 || 0.0858883613 - 0.161150332 i
|-
| 9 || 0.895706001
|-
| 10 || 0.063917354 - 0.131703889 i
|-
| 11 || 0.895283186
|-
| 12 || 0.0505164733 - 0.111197646 i
|-
| 13 || 0.895040036
|-
| 14 || 0.0415832749 - 0.0961519693 i
|-
| 15 || 0.894887492
|-
| 16 || 0.0352431027 - 0.084662092 i
|-
| 17 || 0.89478555
|-
| 18 || 0.0305294065 - 0.0756092279 i
|-
| 19 || 0.894714073
|-
| 20 || 0.0268975698 - 0.0682964386 i
|-
| 21 || 0.894662029
|-
| 22 || 0.0240191066 - 0.0622681185 i
|-
| 23 || 0.894622962
|-
| 24 || 0.0216849331 - 0.0572143342 i
|-
| 25 || 0.89459289
|}
So we have when x is odd, the number is real and converges to 2/sqrt 5. This is rather interesting as it shows that i is not just the solution to the square root of -1, but also the solution to non-integar Fibonacci numbers, which keeps me wondering what the ith Fibonacci root is? I tried using the obove Fibonacci root function wich leaves 2 + 3.2642513 i when you assume i to be odd or 1 + 3.2642513 i when you assume i to be even, with 3.2642513 aparently being what the rank-1 [[Grothendieck constant]] is at most in the tripartite graph G (whatever that means), but when you enter either of those numbers into ((1+sqrt 5)/2)^(1/x)-((-1)^(1/x)/((1+sqrt 5)/2)^(1/x))))/sqrt 5 you do not get i, which suggests the i is neither odd nor even.
Any help will be greatly appreciated. Thanks. [[User:Robo37|Robo37]] ([[User talk:Robo37|talk]]) 11:15, 31 July 2011 (UTC)
:That should be ((1+sqrt 5)/2)^(1/x)-((-1)^(1/x)/(( '''−'''1 +sqrt 5)/2)^(1/x))))/sqrt 5. Your biggest problem when trying to extend the Binet formula to non-integer exponents is that when ''x'' is not an integer, (-1)^''x'' is not well defined. For example, in the second line of your table, you have used the value ''i'' for (-1)^(1/2) rather than -''i'' - this is an arbitrary choice. It is equally valid to say that FR(1/2) could be 0.568864481 '''+''' 0.351577584 i. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 08:03, 9 August 2011 (UTC)
::Thanks for the help, but there must be some kind of value of x when ((1+sqrt 5)/2)^(x)-((-1)^(x)/((1+sqrt 5)/2)^(x))))/sqrt 5=i, right?... is there any way to compute this? [[User:Robo37|Robo37]] ([[User talk:Robo37|talk]]) 10:51, 11 August 2011 (UTC)
::Your function seems to get me to -18.1040617 + 24.5918696 i instead of i for some reason. [[User:Robo37|Robo37]] ([[User talk:Robo37|talk]]) 11:02, 11 August 2011 (UTC)
== Example of Implementation in Programming ==
Seeing as calculation of the Fibonacci Number is a fundamental problem in recursive computer programming, I would find it only appropriate that this page contain at least one example implementation. I feel that many people looking into the Fibonacci Sequence may have some computer background and a quick topic containing some basic code to calculate it, ideally in a functional programming language, would be of great use to them. I would be more than happy to write this up and add it, but as somewhat of a newcomer to the contribution side of Wikipedia I felt it appropriate that I gauge people's feelings on this matter first. Thoughts? <small><span class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Swat510|Swat510]] ([[User talk:Swat510|talk]] • [[Special:Contributions/Swat510|contribs]]) 07:00, 6 August 2011 (UTC)</span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->
{{wikibooks|Fibonacci number program}}
:Thanks for coming here first. [[Fibonacci number#External links]] has the box to the right with a link to a page which was originally a Wikipedia article at [[Fibonacci number program]], but it was transwikied to Wikibooks at [[Wikipedia:Articles for deletion/Fibonacci number program (2 nomination)]]. There are still several examples at [[Recursion (computer science)#Fibonacci]] where it seems more appropriate for the purpose you mention. I have added a link to [[Fibonacci number#See also]]. This seems sufficient for this article which is not suppsed to be about programming. [[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 14:02, 6 August 2011 (UTC)
::Ah, nifty. Didn't see that before. I agree the Recursion topic makes more sense.
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