#### Traditional Formula

There is a well-known formula for the Fibonacci numbers

where

is the golden ratio.

It turns out that there is a way to find for when is not an integer, but the values are complex rather than real.

Mathematics — Algorithms — Version Control

There is a well-known formula for the Fibonacci numbers

where

is the golden ratio.

It turns out that there is a way to find for when is not an integer, but the values are complex rather than real.

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As the article on sums of powers was rather long and dense, I thought that it would be worth giving a summary of the main results separately.

I will also show the formulae in action with a worked example.

In the main article, I show that

This formula is essentially a polynomial of rising factorial powers.

Perhaps the most important and useful formulae from the main article are

and

In the conventional approach to summing powers, that is, finding a polynomial expression for , the coefficients that arise seem to have no pattern. It had always seemed to me that it ought not to be hard to find such expressions with an elementary approach.

There is a well-known formula for the Fibonacci numbers

where

However, I was surprised to find that there are also product formulae involving trigonometric functions.

I noticed that the formulae for the sizes of the surface (or rather, boundary) and volume (interior) of circles, spheres and hyperspheres seem to have discontinuities (or ‘jumps’) in the index of .

However, this is only a result of simplifying a uniform formula for particular cases.

I recently encountered a collection of related articles regarding an amazing explicit enumeration of the non-negative rational numbers. Whilst experimenting with the mathematical constructs described, I made some discoveries of my own (which I doubt are new, but I have not found them described elsewhere).

Below, are the references and links to those articles. They’re all worth a read.

I’ve now uploaded my **Subversion Configuration Suite** (svn-meta) repository to Google Code so that the files are publicly available for anyone to use.

There are many notions of average in mathematics and statistics. Well-known are the *mean*, *median* and *mode*.

Also well-known, amongst the means, are the *arithmetic mean* (A), *geometric mean* (G), *harmonic mean* (H) and *quadratic mean* or *root mean squared* (Q or RMS). Recently, I have become interested in the notion of a *generalised mean* (or *generalized mean*) over positive (non-negative and non-zero) real numbers .

In my post on hypertetrahedral polytopic roots, I gave a few formulae for functions such as the *triangular root*.

Here I give some very similar formulae for some *falling* and *rising roots*.