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