Now you know the secret behind this trick! But this is 1, which is not very interesting,so let's have a look at their Fib(n) factors (since n is a factor of 2n also).Here's a table where F(2n)=kF(n) and we find k for the first few values of n: If we look at the Fibonacci numbers in the even positions (even index numbers)that is Fib(2n),they will all be divisible by Fib(2). For example, since 4 is a factor of 8then Fib(4)=3 is a factor of Fib(8)=24. Lucas Factors of Fibonacci NumbersWhen we began looking at properties of the Fibonacci numbers, we first examined Factors of Fibonacci Numbersand found that if an index number n is a factor of another number m, then the Fibonacci numbers with n and m as index numbersare also factors. Click on the icon here and wherever you see it on this page to go to the online Fibonacci and LucasNumbers Calculator page (in a separate window). These results are shown altogether with many otherson the Fibonacci and Golden RatioFormulae page.Ģ, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843. the name the Fibonacci Numbers,found a similar series occurs often when he was investigating Fibonacci number patterns: The Lucas seriesThe French mathematician, Edouard Lucas (1842-1891), who gave theseries of numbers 0, 1, 1, 2, 3, 5, 8, 13. Yes! No matter what values we start with, positive or negative,the ratio of two neighbouring terms will either be Phi or else -phiĢ, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843. Does it look as (oh dear, I feel a pun coming on: Lucas ) if all the series, no matter what starting values we choose, eventually have successive terms whose ratio is Phi? We know that for the Fibonacci series, the ratio gets closer and closer to Phi = ( 5+1)/2. Investigate what happens to the ratio of successive terms in the series of the earlier questions.Try some other starting values of your own.What if we started with 2 and 1 so that F 0 = 2 and F 1 = 1? Does this become part of the FIbonacci series too? The next two simplest numbers are 2 and 1. What other starting values give the same series as the previous two questions?Īll of which we recognise as (part of) the Fibonacci series after a few terms.What if we started with 2 and 3 so that F 0 = 2 and F 1 = 3?.The calculator iconindicates an interactive calculator in that section. The icon means there are You do the maths.investigations at the end of that section. The followingpage generalises further by taking any two starting values. On this page we examinesome of the interesting properties of the Lucas numbers themselvesas well as looking at its close relationship with the Fibonacci numbers. Edouard Lucas (1842-1891) (who gave the name"Fibonacci Numbers" to the series written about by Leonardo of Pisa) studiedthis second series of numbers: 2, 1, 3, 4, 7, 11, 18. The Lucas Numbers We have seen in earlier pages that there is another series quite similar to theFibonacci series that often occurs when working with the Fibonacci series. Please go to the Preferences for this browser and enable it if you want to use thecalculators, then Reload this page. The calculators on this page require JavaScript but you appear to have switched JavaScript off(it is disabled).
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