Talk:Fibonacci sequence/Archive 3

Archive 1

Archive 2

Archive 3

Archive 4

Citation to Binet's vs. Abraham de Moivre's formula

In paragraph Fibonacci_number#Closed-form_expression citation is needed for disambiguation that closed-form formula was introduces by Abraham de Moivre and not Jacques Philippe Marie Binet. It can be found in the book The_Art_of_Computer_Programming and I think this book should be cited. — Preceding unsigned comment added by Milikicn (talk • contribs) 18:30, 19 August 2011 (UTC)

Simple is best

To initially demonstrate the relationship between the Fibonacci sequence and the Golden Ratio, the Kepler solution is clearly the best. It is the simplest, clearest and most obvious therefore the most elegant solution. The other solutions are definitely worthy of mention but they are needlessly complex answers where a direct answer to a very simple question is already available. The Kepler solution should be the first listed followed by the Binet. Wading through the Binet solution only to find the obvious and to the point Kepler solution leads the reader to conclude that he has stumbled upon an Asperger's self stroking fest rather than an encyclopedia.74.178.137.190 (talk) 11:00, 4 September 2011 (UTC)

Editors here are unlikely to take your suggestions seriously if you cannot express them without throwing in gratuitous playground insults. Gandalf61 (talk) 12:47, 4 September 2011 (UTC)

Identities and combinatorial interpretations

There are two problems with the beginning of the "Identities" section. (1) The first sentence of this section asserts that "Most identities involving Fibonacci numbers draw from combinatorial arguments." This statement sounds subjective; unless reinforced by strong evidence I would remove it. In any case it's irrelevant to the statement of identities. (2) The first identity cannot be proved, as it is the definition. The proper way to handle it is to prove the "interpretation" given (without proof) in the previous section. That should be in a separate section on "Combinatorial interpretations of the Fibonacci numbers". Zaslav (talk) 01:29, 24 October 2011 (UTC)

1/89

The linked page misleadingly suggests that a certain Cody Birsner discovered the relationship between the series and the fraction, whereas it had been known for a considerable time before. Perhaps it would be better to link to another page, e.g. http://www.goldennumber.net/Number89.htm or http://www.fibonacci.name/1-89.html or http://www.mathpages.com/home/kmath108.htm Dadge (talk) 20:50, 31 December 2011 (UTC)

I agree. Thanks for pointing this out. I changed it to:

Köhler, Günter (1985). "Generating functions of Fibonacci-like sequences and decimal expansions of some fractions" (PDF). The Fibonacci Quarterly. 23 (1): 29–35. Retrieved December 31, 2011. {{cite journal}}: Unknown parameter |month= ignored (help)

which in turn cites some earlier papers from FQ. —Mark Dominus (talk) 22:04, 31 December 2011 (UTC)

add some formulas and proofs

put the even,odd,odd,even pattern on the article.and proofs.and before you do this:is there a pattern like this?yes or no and why?John kaiser (talk) 06:08, 30 January 2012 (UTC)

This pattern is mentioned in the section headed "Divisibility properties" and is described more generally in our article on Pisano periods. Gandalf61 (talk) 08:53, 30 January 2012 (UTC)

Recognizing Fibonacci Number

According to the Fibonacci number article "a positive integer $z$ is a Fibonacci number if and only if one of $5z^{2}+4$ or $5z^{2}-4$ is a perfect square."

However, the statement $5z^{2}+4=25^{2}$ or $5z^{2}-4=25^{2}$ does not imply that $z$ is a Fibonacci number. — Preceding unsigned comment added by Yonizilpa (talk • contribs) 19:10, 30 January 2012 (UTC)

You are starting at the wrong end. If z is an integer such that $5z^{2}+4$ or $5z^{2}-4$ is a perfect square then z is a Fibonacci number. So 5x1^2+4 =9, 5x2^2-4=16 and 5x3^2+4=49. It doesn't say there is a Fibonacci number corresponding to every square. Gandalf61 (talk) 19:31, 30 January 2012 (UTC)

In fact, the only squares that give Fibonacci numbers in the reverse direction are the squares of Lucas numbers. 1,2,3,4 and 7 are Lucas numbers, but 5 and 6 are not, so we have

${\frac {1^{2}+4}{5}}=1=1^{2}$

${\frac {2^{2}-4}{5}}=0=0^{2}$

${\frac {3^{2}-4}{5}}=1=1^{2}$

${\frac {4^{2}+4}{5}}=4=2^{2}$

${\frac {7^{2}-4}{5}}=9=3^{2}$

which are all squares of Fibonacci numbers, but ${\frac {5^{2}\pm 4}{5}}\,$ and ${\frac {6^{2}\pm 4}{5}}\,$ are not squares of Fibonacci numbers. Gandalf61 (talk) 09:04, 31 January 2012 (UTC)

Thanks for the explanation I now realize my mistake. — Preceding unsigned comment added by Yonizilpa (talk • contribs) 13:10, 11 February 2012 (UTC)

A pyramid relating the golden ratio and the tetrahedron

If you take a regular tetrahedron and truncate(cut) it so that you keep the three original 60degree angles at one vertex but change the three lengths from that vertex to any three successive terms of the Fibonacci series then the base of the new pyramid will be the two internal diagonals of a pentagon and the corresponding side.The face with the corresponding side has the other sides Fib(n)and Fib(n+1),the angle opposite Fib(n) is 37.76124degrees and opposite Fib(n+1) is 82.23876degrees. One face of the pyramid with one of the internal diagonals as a side has other sides Fib(n+1) and Fib(n+2) and is similar to the previously mentioned face. The third face having the other internal diagonal as a side has other sides Fib(n)-angle opposite being 22.23876degrees- and Fib(n+2)-angle opposite being 97.76124degrees. Sabastianblak (talk) 23:20, 12 February 2012 (UTC)Bradley J. Grantham February 12,2012

The fibonacci numbers used for multiplication

The Fibonacci numbers are meso-American multiplication method. It may use any type of numbers I think.... — Preceding unsigned comment added by 203.126.140.131 (talk) 01:26, 29 March 2012 (UTC)

Nobel Prize ?

Excuse me but, i win a nobel prize if i say that the fibonacci phenomenon is due to rotation of earth "plus" growth factor ???

for example... for the spiral of sunflowers, and sea animal shells, just draw a straight line...very slowly...maybe following the sun light... and the rotation of earth make (plus growth factor) it become a spiral... does anybody before me understand it ???

thank you.

M.G. — Preceding unsigned comment added by 79.25.124.250 (talk) 15:39, 11 June 2012 (UTC)

See golden spiral. Gandalf61 (talk) 15:47, 11 June 2012 (UTC)

ty i read..but..no mention to relate it to the rotation of earth? (or taking the sun as a polar star)? — Preceding unsigned comment added by 79.25.124.250 (talk) 16:09, 11 June 2012 (UTC)

Well, you have a hypothesis, which is that phyllotaxis is caused by the movement of the sun across the sky. But that is just the first part of the scientific method. Next you need to make a prediction - for example, if the sun did not move across the sky, then plants would not show phyllotaxis. Then you need to devise and carry out an experiment that tests your prediction - for example, you could grow plants under artificial sun lamps so that the light under which they grow comes from a constant direction. If you did all this, and got a positive result from the experiment, then it would be interesting, but I really doubt it would qualify for a Nobel Prize. Gandalf61 (talk) 08:56, 12 June 2012 (UTC)

aaaa!! ty Gandalf61!! :) mmm i already have a certain idea..:) but i will make some experiment anyway. ty so much for your interest on my topic anyway! thank you. and...if i suppose is due to the earth magnetic field..? kinda like make grow a broccoflower inside a solenoid? i will try.

-mmm..nice reading Phyllotaxis..it comes out that golden ratio have something to do with pentagon, and pentagon with dodecahedron, a platonic solid.(and the that-time-believed shape of universe). Bye. — Preceding unsigned comment added by 79.11.127.63 (talk) 18:34, 12 June 2012 (UTC)

first list of fibonacci numbers

in the image /media/math/c/a/b/cabe91689f6a1af616ace02827c6e89c.png shouldn't the 7 be an 8? — Preceding unsigned comment added by JaysnArr (talk • contribs) 18:55, 5 October 2012 (UTC)

It says 8 for me but I see 7 in /media/math/7/3/8/73824f3b44e0a4920a70ae1ff1820fcd.png. The article was recently changed to a wrong sequence with 7. It was fixed 10 minutes later.[1] PrimeHunter (talk) 19:02, 5 October 2012 (UTC)

Initial Fibonacci numbers

The original Fibonacci numbers began 1, 1, as everyone knows. In modern times mathematicians have often found it useful to start with 0, 1, 1, but often they have not found that useful and still begin with 1, 1. The article ought not to be prescribing usage to mathematicians, and I have revised the introduction accordingly. Some time ago, a person objected to any use of the beginning 1, 1, so I provide a reference for the 1, 1 sequence:

Matthias Beck and Ross Geoghegan, The Art of Proof: Basic Training for Deeper Mathematics. New York: Springer, 2010.

Zaslav (talk) 06:46, 9 October 2012 (UTC)

Addendum (not merely beating a dead horse, I'm sorry to say): In applications of Fibonacci numbers in science, the initial numbers seem to be 1, 1, not 0, 1, 1 (0 being inappropriate for them). That is certainly true in biology. Users of Fibonacci numbers in science (or anywhere else), not only professional mathematicians, ought to be taken account of in deciding what "the" initial numbers are. Zaslav (talk) 06:16, 23 October 2012 (UTC)

Starting with 0 makes both the generating function and the closed-form expression in terms of the golden ratio nicer.

It's not really an issue, though, since the mathematics and computer science references I see tend to refer to F_0=0, F_1=1, F_2=1, F_3=2, etc. That makes terms 1,2 of the sequence 1,1 for the life sciences, and "term 0" is there for those who want to start with 0. Husoski (talk) 18:45, 25 January 2013 (UTC)