phi2cfrac: Function to Calculate Phi And Powers of Phi In Continued Fraction Form.

Description

This function generates the continued fraction form of the "golden ratio", phi^N for integer powers N.

Usage

phi2cfrac( nterms = 10, exponent = 1,  ...)

Value

The continued fraction denominators are provided in denom. The inputs nterms and exponent are echoed back for reference.

Arguments

nterms: How many denominators to calculate.
exponent: An positive integer indicating the power of phi desired. The default is 1.
...: Reserved for future use.

Author

Carl Witthoft, carl@witthoft.com

Details

The 'golden ratio' , equal to (1 + sqrt(5))/2, is the ratio of two sides x < y of a rectangle such that, by removing a square of side x, the remaining rectangle has the same ratio.

It turns out, in one of those mathematical curiosities, the denominators of the continued fraction form of phi are all equal to one. Some people use this to state, humorously, that this makes phi "the most irrational irrational number." It also happens that the continued fraction form for powers of phi consist of Lucas Numbers (see References).

References

https://en.wikipedia.org/wiki/Lucas_number https://en.wikipedia.org/wiki/Golden_ratio

Examples

Run this code

phi2cfrac(nterms = 10)
phi2cfrac(exponent = 3)
foop <- phi2cfrac(nterms = 20)
cfrac2num(denom  = foop$denom)
# compare with:
library(Rmpfr)
(1 + sqrt(mpfr(5,1000)))/2

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