## September 29, 2010

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Very cool.

Here's an easy 3-liner that I'm sure could simply be wrapped into a function.

fv<-c(0,1)
sapply(3:20, function(i) fv[i]<<-fv[i-1]+fv[i-2])
fv[-1]/fv[-length(fv)

Which, oh, that could be compressed into 2 lines (you have to initialize the vector)

fv<-c()
sapply(1:20, function(i) if(i<3){fv[i] <<- 1}else{fv[i]<<-fv[i-1]+fv[i-2]})[-1]/fv[-length(fv)]

Via John Myles White, here's an amazingly elegant way of deriving the Golden Mean from its continued fraction representation using R's Reduce function:


> cfrac <- function(x) Reduce(function(u, v) u + 1 / v, x, right = TRUE)
> cfrac(rep(1,10))
[1] 1.618182


cfrac(rep(1,n)) better approximates the Golden Ratio as n gets larger.

It turns out that you can understand why the ratio is the golden mean if you cast the sequence in terms of iteration of a certain matrix:
http://blogs.sas.com/iml/index.php?/archives/22-Matrices,-Eigenvalues,-Fibonacci,-and-the-Golden-Ratio.html

This is very exciting and I like your template as well. Keep it up.

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