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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>
Line 415:
return fib(n-1)+fib(n-2);
}
</syntaxhighlight>
== MathWorld Fibonacci article should be linked ==
Line 422:
:There is a link to the MathWorld article under Notes, reference 15. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 09:23, 12 May 2009 (UTC)
::I know. I say there should be one in the external links, even so. Best wishes, [[Special:Contributions/75.45.125.91|75.45.125.91]] ([[User talk:75.45.125.91|talk]]) 15:29, 12 May 2009 (UTC)[[User:Richard L. Peterson|Rich]] ([[User talk:Richard L. Peterson|talk]]) 15:31, 12 May 2009 (UTC)
== True or False?? ==
True or false: It has been proven that 149 is the largest prime Tribonacci number.
:False, there are larger prime Tribonacci numbers, e.g. 19341322569415713958901, 15762679542071167858843489 and 145082467753351661438130501937754420584096000083183992629 {{OEIS|id=A092836}} [[User:Trewal|Trewal]] ([[User talk:Trewal|talk]]) 12:17, 2 June 2009 (UTC)
== somebody. pls translate this to alphabets ==
1.144.17711_1.5.1_(2)_8.144.987.4181_1.121393.1.377
5.17711.1-5.17711.1_610.1.144_6765.8.377.2.8.233.55.34
2.987.233.8.34_46368.1.10946_4181.8.610.5.1.610.21
34.1.34.1
10946.55.377
1.144.17711_6765.8.2.8.610.1.4181.610.121393.1_
6765.1.121393.1.610.21.144.1.610_144.1.17711
377.17711.610.21.144.55.610_1.144.17711_10946.8.4181.233.1.233.17711_
8.377.987
_10946.1.1597.55
89.1.17711.34_5.1.233.1.377_34.1.10946.55,
1.144.17711_6765.1.121393.1.610.21.144.1.610_144.1.17711.
377.1.1.13.144.1.610_1.144.17711_144.1.233.1.17711_
144.1.10946.1.-144.1.10946.1_1.144.17711
46368.1.10946_144.1.17711_10946.8.4181.233.17711.144.1 -end-
6765.1.1597.1_2.987.233.8.34_10946.8.144.1_5.55.1_
377.377.21_1597.1.610.5.1.55 <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/121.121.67.100|121.121.67.100]] ([[User talk:121.121.67.100|talk]]) 08:58, 9 June 2009 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->
:It's off-topic, but I enjoyed the puzzle. It seems to be some sort of message in Malay (according to running Google's language detection on my result), in a code that maps letters of the alphabet to Fibonacci numbers (A=1, B=2, C=3, D=5...). If I have the mapping right, it says:
::AKU ADA (B) EKOS AYAM DUA-DUA NAK SEMBELIH BOLEH WAT SENDANG HAHA
::TIM AKU SEBENASNYA SAYANGKAN KAU MUNGKIN AKU TESLALU EMO TAQI JAUH DALAM HATI, AKU SAYANGKAN KAU MAAFKAN AKU KALAU KATA-KATA AKU WAT KAU TESLUKA -end-
::SAQA BOLEH TEKA DIA MMG QANDAI
:And now I'm curious what it means. [[User:Rspeer|rspεεr]] ([[User talk:Rspeer|talk]]) 17:09, 14 June 2009 (UTC)
== Fibonicci.co.uk/number-sequences removed discussion. ==
Hello,
I Got a message that i should discuss the removal of the link.
I don't see why this link does not contribute to the Fibonacci article.
1) The website's name www.fibonicci.co.uk in itself is a referral to the fibonacci sequence
2) Also the logo, of a nautilus shape, is based on the fibonacci sequence
3) The test them self, the medium and the hard one, contains several sequences based that mimic the fibonacci secuence.
4) As far as I looked, the structure of all tests are build up in 8, 13, and 21 questions for the easy, medium and hard tests. Same as fibonacci sequence.
So for me its unclear why this link doesn't belong in the fibonacci article. Especially when compared to the 1st link: On-Line Encyclopedia of Integer Sequences.
This has almost nothing to do with the fibonacci sequence. The only relationship being that it is a sequence.
I havent measured the outline of the site, but wouldnt be surprised if somewhere also the golden ratio is applied. At first glance I can understand that maybe the site doesn't belong to the article, but when looked a bit deeper and taken into account all the referals to fibonacci, than in my opinion this is a valid link.
http://www.fibonicci.co.uk/number-sequences
Just my two cents..
[[User:Exodian|Exodian]] ([[User talk:Exodian|talk]]) 19:45, 11 June 2009 (UTC)
:See [[WP:EL]]. — [[User:Miym|Miym]] ([[User talk:Miym|talk]]) 20:36, 11 June 2009 (UTC)
== 1/89 = Fibonacci Sequence ==
Interesting fact: http://www.geom.uiuc.edu/~rminer/1over89<br />
Maybe this could be added to the article? --[[Special:Contributions/41.204.111.27|41.204.111.27]] ([[User talk:41.204.111.27|talk]]) 00:27, 12 June 2009 (UTC)
:It is already mention in the section ''Power series'', but I have added your link as a source. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 14:07, 12 June 2009 (UTC)
::Indeed, many editors get confused about these two articles; the remarkable number 1/89 is already in [[Fibonacci_number#Power_series]], with a ref, but done up in a way that make the relationship described above very hard to see, so that might be a good thing to improve on. [[User:Dicklyon|Dicklyon]] ([[User talk:Dicklyon|talk]]) 03:37, 25 September 2009 (UTC)
== fibonacci when did he invent it and why ==
who what when where and why i need these answers <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/94.0.158.49|94.0.158.49]] ([[User talk:94.0.158.49|talk]]) 15:14, 29 September 2009 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->
:Who, what and when are all in the first few lines of this article. For where and why, see our article on [[Liber Abaci]]. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 15:33, 29 September 2009 (UTC)
== Simple Cell division rate as explanation for Fibonacci numbers and for golden mean in nature. ==
[http://deoxy.org/forum/showflat.pl?Cat=1&Board=vox&Number=55595&page=0&view=collapsed&sb=5]
On the first topic thread there is the proposal, and linked simple diagram.
Then follows open discussion, which at some point turns into computed tests to disprove or prove the explanation. Thus far no fault seems to have been found in it. Seems deceptively simple...
In a nutshell: an organism, like cell (or even self replicating bubble ?) is born. Then it exists and grows. Then it divides, gives birth to another organism. While the new "child" organism is still gathering energy and growing, the original orgnism splits again. Then there is three. Next the elder child organism and parent organism split while youngest one still grows, and there is 5 of them, and it goes on and on and seems to give the fibonacci number each time, as well as the golden proportion in shapes and volumes, even if some of the organisms get wrecked at some generations, or starting amount of organisms is varied.
Direct link to the cell division diagram only, without the discussions, commentary and testing: [http://koti.mbnet.fi/maxt/oddsnends/fibo.gif]
My sources have been... I just started to calculate it as visual organic division, without numbers. I think I was trying to understand development of organic spatial forms. No other sources, except that I did not seem to find quite this simple explanation that gives reproducible results, in wikipedia or elsewhere in the Net. At least not for now.
It is so very simple that it may be wondered why I post it... and the answer is, because I have not yet seen it elsewhere, only more complex models with more parameters, and things like rabbits and bee communities ratios of young queens.
--[[User:MaxTperson|MaxTperson]] ([[User talk:MaxTperson|talk]]) 14:49, 12 January 2010 (UTC)
== 13 Popular Culture [[http://en.wikipedia.org/wiki/Fibonacci_number#Popular_culture]] ==
No mention here of the popular culture that exists in the [[Stock]] Trading business, [[Economics]], [[Mysticism]] and other parts of the culture that are continually on the look out for Fibonacci Numbers in any series - it is because it is a "Natural" Series. Just look at the external links to see just how wide spread and deeply influential in popular culture this series is.
Good point! Also Bartok and other composers! [[Special:Contributions/75.48.16.182|75.48.16.182]] ([[User talk:75.48.16.182|talk]]) 20:09, 2 October 2010 (UTC)
== F(x) for fabonci ==
f(x) = exp(0.481219*x - 0.8049) + 0.00114292*x^2 - -0.355997^x - 0.0112628*x
or a bit more complex
f(x) = exp(0.481219*x - 0.804903) + 0.00160452*x^2 + 0.0126129*log(x) - -0.364351^x - 0.00160452*x*log(x) - 0.014535*x - 0.00226154
(amazing results pretty close isn't it ??) <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/82.217.115.160|82.217.115.160]] ([[User talk:82.217.115.160|talk]]) 17:12, 3 April 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->
Weird, I did not post this... starting from " F(x) for fabonci " is not my text or opinions at all. Why is it on my signature ? <small><span class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:MaxTperson|MaxTperson]] ([[User talk:MaxTperson|talk]] • [[Special:Contributions/MaxTperson|contribs]]) 22:20, 11 October 2010 (UTC)</span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->
:Probably because whoever did post it put it between your post and its signature. I've moved your sig up; please check whether it is now in the right place. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 22:25, 11 October 2010 (UTC)
== Incorrect Equivalence ==
See my comments on [http://stackoverflow.com/questions/2432669/test-if-a-number-is-fibonacci/2432699#2432699]. --[[User:Andreas Rejbrand|Andreas Rejbrand]] ([[User talk:Andreas Rejbrand|talk]]) 22:47, 12 May 2010 (UTC)
== Easy way but take time ==
The pattern is like this <br>
0 + 1 = 1<br>
1 + 1 = 2<br>
2 + 1 = 3<br>
3 + 2 = 5<br>
5 + 3 = 8<br>
8 + 5 = 13<br>
13 + 8 = 21<br>
21 + 13 = 34<br>
34 + 21 = 55<br>
How many year you finish to the numbers 100? --[[User:Samit Boonyaruk|Samit Boonyaruk]] ([[User talk:Samit Boonyaruk|talk]]) 14:29, 24 August 2010 (UTC)
:It gets too large for a regular calculator ({{#expr:((1+5^.5)/2)^100/5^.5}}), but with pen and paper it may be nice for a rainy day.--[[User:Patrick|Patrick]] ([[User talk:Patrick|talk]]) 14:50, 24 August 2010 (UTC)
== A matter of style ==
In the article we have: "the (n + 2)'''nd''' Fibonacci number". Why "nd"? Is it because it's preceded by a "2"? To stop people saying "n plus tooth"? I think "th" is better.
But the article isn't consistent: lower down we have: "we consider the (n + 1)'''th''' summand". To be consistent it would say: "(n + 1)st". I think "th" is better.
Show of hands for an edit? Change "(n + 2)nd" to "(n + 2)th"?
[[User:AstroWiki|R L Lacchin (Gloucester, UK)]] ([[User talk:AstroWiki|talk]]) 14:05, 28 October 2010 (UTC)
I think the two-letter suffix is "stretching" the usage too far. I'd say that a more-elaborate construction is better.
(Fwiw, I was employed as an editor for a couple of years, and had very few copy-edit changes to what I typed.) [[User:Nikevich|Nikevich]] ([[User talk:Nikevich|talk]]) 06:07, 29 November 2010 (UTC)
== Contradicting Connotations? ==
In the "In nature" section, there seem to be some balancing problems in the first paragraph:
"Fibonacci sequences appear in biological settings, [various examples of the sequence in nature]. In addition, numerous poorly substantiated claims of Fibonacci numbers or golden sections in nature are found in popular sources..."
The use of "In addition" and "poorly" is slightly awkward and contradictory. I'd recommend something like "There are numerous claims of Fibonacci numbers or golden sections in nature, although many are poorly substantiated." Thoughts? [[Special:Contributions/174.91.190.188|174.91.190.188]] ([[User talk:174.91.190.188|talk]]) 03:25, 13 December 2010 (UTC)
:Okay with me. —[[User:Dominus|Mark Dominus]] ([[User talk:Dominus|talk]]) 04:19, 13 December 2010 (UTC)
== Systematic bias in revert by David Eppstein ==
This is related to the revert by David Eppstein and the prejudiced remarks with which a cited
fact was reverted, along with other edits.
The name [[Gopala-Hemachandra number]] is gaining currency as a general form of the Fibonacci
number. Two mathematics papers, one from the J of Number theory, were cited to show
how this term is used.
Whether the term is "a trivial extension" is not an issue on Wikipedia. The fact is that
it is being used, and we need to report it here.
If you had argued about the citations, said that they are non-authoritative, showed other
evidence that these are not coherent or otherwise, I would have understood this. On wikipedia
it doesn't suffice to say that the editor has an Indian-sounding
name and hence this is Nationalistic edit.
In fact, there is stronger evidence for [[Wikipedia:Systemic bias]] (eurocentric bias) in
this revert than any factual stance.
The revert also had other edits other than this issue, such as the incorrect "disputed" characterization.
Please re-edit the text, if needed, do not revert.
At the very least, such issues should be discussed here before reverting edits.
I am reverting the revert and will expect a civilized response on this page rather than a revert war. [[User:Mukerjee|mukerjee]] ([[User talk:Mukerjee|talk]]) 06:41, 26 March 2011 (UTC)
:The phrase "Gopala-Hemachandra number" has exactly zero hits in Google scholar. And if it had two hits, that would still be a tiny number compared to the vast literature on Fibonacci numbers. As for the disputed part, please read the paragraph in the "Origins" section, which clearly outlines the dispute among scholars about the timing of the discovery in India of the Fibonacci numbers, and clearly contradicts your poorly-sourced edits in the lead section.. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 06:49, 26 March 2011 (UTC)
::I get four hits, including the Basu-Prasad paper, the Thomas paper, and two more. Is gscholar geography-biased? [[User:Mukerjee|mukerjee]] ([[User talk:Mukerjee|talk]]) 07:16, 26 March 2011 (UTC)
:I reviewed the first revert carefully, as it seemed that it might be better to include this material in the article. The more I looked at it though, the more it became clear that a much more complete and well referenced treatment of the contributions of these two guys was already in the article. There's no need to put this obscure and controversial bit into the lead as well. [[User:Dicklyon|Dicklyon]] ([[User talk:Dicklyon|talk]]) 06:54, 26 March 2011 (UTC)
:David:
:You make the point
::nationalist claims of priority that are sourceable but also disputed in the literature.
:Pls cite a source that "disputes" the Indian prior claim. I can't find any and would be happy to have some. The three articles cited in the article all dispute whether the Indian work dates from 400BC or 700BC, no one argues against any priority.
:But this is an aside. The main point is that there are references using this name.
::I am attaching some segments of the paper. It is available through elsevier at http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6WKD-4YT7KMP-1-1&_cdi=6904&_user=489944&_pii=S0022314X10000533&_origin=gateway&_coverDate=09%2F30%2F2010&_sk=998699990&view=c&wchp=dGLzVlz-zSkzk&md5=f2a86d5e45ce40cf5875f53017c9befb&ie=/sdarticle.pdf
::: Here sre some extracts from the Basu-Prasad paper:
:::Journal of Number Theory 130 (2010) 1925–1931
::: Long range variations on the Fibonacci universal code
::: Manjusri Basu, Bandhu Prasad Received 15 September 2009 Revised 6 January 2010 Available online 8 April 2010 Communicated by David Goss
:::Abstract
:::Fibonacci coding is based on Fibonacci numbers and was defined by Apostolico nd Fraenkel (1987) [1]. Fibonacci numbers are generated by the recurrence relation Fi = Fi−1 + Fi−2 ∀i >= 2 with initial terms F0 = 1, F1 = 1. Variations on the Fibonacci coding are used in source coding as well as in cryptography. this paper, we have extended the table given by Thomas [8] We have found that there is no Gopala–Hemachandra code for a particular positive integer n and for a particular value of a ∈ Z . We conclude that for n = 1, 2, 3, 4, Gopala–Hemachandra code exists for a = −2, −3, . . . , −20. Also, for 1 � n � 100, there is at most m consecutive not available (N/A) Gopala–Hemachandra code in GH−(4+m) column where 1 � m � 16. And, for 1 � n � 100, as m increases the availability of Gopala–Hemachandra code decreases in GH−(4+m) column where 1 � m � 16.
:::... And section 2 has:
:::2. Gopala–Hemachandra (GH) sequence and codes
:::A variation to the Fibonacci sequence is the more general GH sequence [6] {a, b, a + b, a + 2b, 2a + 3b, 3a + 5b, . . .} for any pair a, b which for the case a = 1, b = 2 represents the Fibonacci numbers [4,5,7] gives historical details of these sequences.
::The paper is available from Elsevier at http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6WKD-4YT7KMP-1-1&_cdi=6904 (or at most univ libraries)[[User:Mukerjee|mukerjee]] ([[User talk:Mukerjee|talk]]) 07:16, 26 March 2011 (UTC)
===Origins section===
I hate edit wars.
re-reading the whole article, I find that the origin section has the sentence, which is perhaps part of what Dicklyon is referring to:
:However, according to Knuth, neither of these works concern the more specific problem of counting patterns with a fixed total duration, for which the answer is the Fibonacci numbers, and for which Knuth cites instead the anonymous Prakrta Paingala (c. 1320 AD), which uses [[Fibonacci coding]] to solve the problem.
But what Knuth says in the relevant part is:
:... consider the set of all sequences of L and S that have exactly m beats. ... there are eactly Fm+1 of them. For example the 21 sequences when m=7 are: [List]
:In this way Indian prosodists were led to discover the Fibonacci sequence, as we have observed in Section 1.2.8
:Moreover, the anonymous author of prAkr.ta paingala (c.1320) discovered elegant algorithms for ranking and unranking w.r.t. m-beat rhythms. p. 50
Thus the statement "neither of these works concern the more specific problem of counting patterns with a fixed total duration, for which the answer is the Fibonacci numbers" appears to be a deliberate misleading the reader. In fact, Knuth says unequivocally in v.1 p.100:
: Before Fibonacci wrote his work, the sequence Fn had already been discussed by Indian scholars, who had long been interested in rhythmic patterns... both Gopala (before 1135AD) and Hemachandra (c.1150) mentioned the numbers 1,2,3,5,8,13,21 ... explicitly. [See P. Singh Historia Math 12 (1985) 229-244] AOP v.1 3d ed. p. 100
I am not sure who inserted this sentence, but if there is evidence of prejudice it is in this sentence, which is also used to make an incorrect claim about what was disputed in the lead.
Would anyone object if I edited the origins section, or is the above text justified? [[User:Mukerjee|mukerjee]] ([[User talk:Mukerjee|talk]]) 09:01, 26 March 2011 (UTC)
:I don't have immediate access to a recent edition of Knuth Vol.I which is what you seem to be quoting; the part about Gopala and Hemachandra certainly isn't in the part of Knuth that is cited in the article. Maybe someone else who does have a recent edition and can check what Mukerjee says would like to weigh in? —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 23:11, 26 March 2011 (UTC)
::there is no problem accessing [[AOP]] - i have also provided the google books links.
::Even if others haven't accessed it yet, it follows wikipedia verifiability norms so I am going ahead and editing in the changes.
:: Pls note that wikipedia [[wikipedia:Editing policy|policy]] suggests: "Please boldly add information to Wikipedia, either by creating new articles or adding to existing articles, and exercise particular caution when considering removing information."
::So pls take added care in removing edits that contain verifiable facts.
::I will next be re-inserting some of the original text in the lead. [[User:Mukerjee|mukerjee]] ([[User talk:Mukerjee|talk]])
== Sunflower image ==
I removed the sunflower image since when I tried to verify the spiral count I kept getting different results. I think the particular sunflower in the image had irregularities that made the spirals too unstable to make the counts meaningful. The other sunflower images I found weren't much better but I did find a chamomile image where I could draw in the spirals. I also removed the seashell image, the connection to the Fibonacci number has been claimed by some authors but my understanding is that this has been debunked.--[[User:RDBury|RDBury]] ([[User talk:RDBury|talk]]) 02:05, 28 April 2011 (UTC)
== First number ==
As the article notes, the Fibonacci sequence is sometimes said to start with 0, sometimes with 1. Strictly speaking, if you are calling this sequence the "Fibonacci" sequence, you should begin with 1 because that is how Fibonacci started the sequence. Might I suggest you start with 1 and note that "some sources start with 0". In addition to historical accuracy, I suspect starting with 1 is far more popular in the literature, despite the OEIS entry. Or perhaps it would require too much rewriting of the article to make it worth the trouble to be historically accurate? --[[User:Seberle|seberle]] ([[User talk:Seberle|talk]]) 06:06, 26 February 2011 (UTC)
I admit this is probably minor, but it still bugs me that starting with zero is not, very strictly speaking, the "Fibonacci" sequence, nor is it the sequence that most readers will come across in other readings. At the very least, could we replace "Some sources omit the initial 0" in the introduction with "Fibonacci originally started his sequence with two 1s instead of 0 and 1, and many still define the sequence that way instead"? The way it's written now makes it look as if starting with zero were the definitive way and starting with two 1s is an infrequent variation. --[[User:Seberle|seberle]] ([[User talk:Seberle|talk]]) 17:12, 27 February 2011 (UTC)
:I don't think Fibonacci called it "the Fibonacci sequence". It was named later, and named objects often deviate from the work of the person they were named after. Including 0 is mathematically important for many of the formulas and should be "our" definition, but we can reformulate the description of the alternative definition. It currently says: "Some sources omit the initial 0, instead beginning the sequence with two 1s". That can indeed give a misleading impression of the history and prevalence. I suggest: "Many sources do not include an initial 0, instead beginning the sequence with two 1s". [[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 17:54, 27 February 2011 (UTC)
:There are two slightly different issues here:
::(A) Should the enumeration of the sequence start with
:::<math>F_0 = 0 \quad\text{and}\quad F_1 = 1\, ,</math>
::or with
:::<math>F_0 = 1 \quad\text{and}\quad F_1 = 1\, .</math>
::(B) Should the quoted sequence of values start 0,1,1,... or start 1,1,2,...
:On (A), it is clear that the enumeration must start with <math>F_0 = 0</math>; if we change this, many of the quoted properties of the sequence do not hold (for example, ''F''<sub>''kn''</sub> is no longer divisibel by ''F''<sub>''n''</sub>). On (B), I have no preference. On either point, I think that the argument "this is not how Fibonacci defined the sequence" does not carry much weight - many mathematical objects and terms have modern definitions that differ widely from those given by their originators. Fibonacci's original formulation may be of historical interest but it should not dictate how we write our article. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 10:39, 28 February 2011 (UTC)
I agree. Clearly the best way to define the Fibonacci sequence is as it is done in Wikipedia. However, we are still dealing with two problems. (This is all rather minor really, a question of wording in the introduction.) (1) The sequence was historically defined as beginning 1, 1, ... This goes beyond how Fibonacci defined the sequence (and yes of course he didn't name it after himself -- he didn't name it all as far as I know) because (2) many sources (and I strongly suspect the majority of popular sources) still begin with 1, 1, ... today; this is not some archaic historical distinction. Therefore the remark "Some sources omit the initial 0", as if starting with 1 was an afterthought, is rather misleading. Though not intentionally stated, the wording gives the impression that those who start with 1 are doing something wrong. PrimeHunter's proposal is an improvement, though it still implies something is being left out, when actually 0 is being added to the original sequence for the reasons Gandalf61 points out. And it would still be nice to include that this is how the sequence was historically defined (and not vice versa). It really shouldn't be hard to come up with a better wording for this one sentence. --[[User:Seberle|seberle]] ([[User talk:Seberle|talk]]) 16:30, 28 February 2011 (UTC)
:The separation into (A) and (B) is good. I agree with the first choice in (A). This should not be too controversial though it isn't done by everyone.
:Here's what I remember from what I've seen in math and science, which may be correct: "the Fibonacci numbers" usually start with 1, 1 and then usually they're called ''F''<sub>1</sub>, ''F''<sub>2</sub>. The math people would generally be happy to include ''F''<sub>0</sub> = 0. Some would like to say ''F''<sub>0</sub> = 1, ''F''<sub>1</sub> = 1. Some experts would say it depends on what suits the context.
:I do think most people who know the sequence would start it 1,1,....
:Here is an argument for not including 0 in "the" (basic) sequence. The number ''f''<sub>''n''</sub> of sequences of 1's and 2's of total length ''n'' starts with ''f''<sub>0</sub> = 1, not 0. The value 0 doesn't have any place in this count. Many other things counted by Fibonacci numbers behave the same way. I don't doubt there are counterexamples; I'm just mentioning that this is one justification, in addition to the original Fibonacci problem, for preferring to start at 1, or at least not preferring to start at 0. In my experience the single best reason for including 0 in the sequence is the attractiveness of working backwards via the recurrence relation.
:My suggestion would be to revise the first few lines to say the sequence is 1,1,2,3,..., then say some prefer 0,1,1,2,3,..., but mathematically the conventional recurrence may start with initial conditions either ''F''<sub>0</sub> = 0, ''F''<sub>1</sub> = 1 or ''F''<sub>1</sub> = 1, ''F''<sub>2</sub> = 1; and furthermore some people displace the sequence and say ''F''<sub>0</sub> = 1, ''F''<sub>1</sub> = 1. This would be true to real life, which would probably be more useful to ordinary users than a single rigid definition. [[User:Zaslav|Zaslav]] ([[User talk:Zaslav|talk]]) 03:06, 26 March 2011 (UTC)
I believe that the history of how the Fibonacci series came about informs this discussion. That history is already in the article under the heading Origins. But Origins should start with the Fibonacci story if we are to retain the word Fibonacci in the title of the entry. The Indian mathematics which developed the same series at an earlier time should appear but not before the Fibonacci history is told. To do this in no way diminishes the Indian achievement - it just places it in the context of an article on the Fibonacci series. Fibonacci himself developed the series to do a job. That job was to calculate how many rabbits you would have after a year if you took as your data the number of kittens rabbits usually have, and how often they are fertile and assuming that on average they have equal numbers of male and female offspring. The problem thus starts with a pair of rabbits. So the series as conceived by Fibonacci starts with 1. (one pair of rabbits) Some might think that 0 should be the first number so that the series can start using the rule ''F''<sub>n</sub> = ''F''<sub>n-1</sub> + ''F''<sub>n-2</sub>. If we use this rule and start as Fibonacci started then we start with 1 and add the number which came before it. No number came before it so add zero. 1+0=1 the second number. And so on. But if we start with zero then the series gets nowhere if we wish to find the second number by the rule because if we start with 0 then the next number must be 0+0=0 and the series gets nowhere.
Even the Indian series starts with a whole number greater than zero. So starting with zero is really not only historically incorrect but conceptually incorrect and it is thus misleading as well.
This leads me to the idea that the article might be renamed The Fibonacci Series or Fibonacci Numbers. In order to recognise the common but historically and conceptually incorrect way of developing the series a footnote or a comment might be made that "some people after Fibonacci used to start and some still start the series with zero" and then the article could go on to comment that this is historically and conceptually correct and misleading.
[[User:Robarc|Robarc]] ([[User talk:Robarc|talk]]) 17:22, 12 June 2011 (UTC)
:In view of the discussion above I am changing the introduction to state that the series begins 1, 1 or 0, 1, depending on personal preference. [[User:Zaslav|Zaslav]] ([[User talk:Zaslav|talk]]) 05:52, 28 June 2011 (UTC)
:It seemed best to allow that the sequence may begin 1,1 or 0,1, and the numbering of the sequence may begin ''F''<sub>0</sub>,''F''<sub>1</sub> in either case or ''F''<sub>1</sub>,''F''<sub>1</sub> in the former case. There is no general agreement about which to use. I stated that in this article 0,1 is used. [[User:Zaslav|Zaslav]] ([[User talk:Zaslav|talk]]) 06:15, 28 June 2011 (UTC)
== Relation to Golden Ratio Information ==
I am not a mathematician, but the following sentence in the information beneath the image in the Golden Ratio section does not properly describe the equations in the image: "The length of the side of one square divided by that of the next smaller square is the golden ratio." Could someone that knows better fix either the image or the text?--[[User:Bainst|Bainst]] ([[User talk:Bainst|talk]]) 19:00, 1 May 2011 (UTC)
: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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