#### 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

A PDF version of the previous post is here.

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.

For quite some time now, I have been looking in books and online for a set of tables with formulae for conversion between various measures of the platonic solids (the regular polyhedra). None quite fitted my requirements, and so I created my own.

My requirements included:

- The formulae should all be of a similar form.
- Where there is a change of dimension, formulae should be given both in terms of the source and the target dimensons.
- No formula should have a surd (root) in the denominator.
- The terms in a surd should have reduced factors. (So, in particular, any integer under a square root should be square-free.)

The set of numbers can be partitioned into two subsets of the same size, such that the two sets have equal sums, sums of squares, sums of cubes, …, up to sums of th powers.

For example, for :

can be partitioned as

so that

and, lastly,

Amazingly, this can be done for any non-negative integer .

This is a link to a historical project of mine, hosted on Albert Gräf’s project page.

QiX is a library for Albert Gräf’s Q programming language adding support for univariate polynomials.

There is full documentation available.

This is a link to a historical project of mine, hosted on Albert Gräf’s project page.

Q+Q is a library for Albert Gräf’s Q programming language adding support for the rational numbers, ℚ.

There is full documentation available.

In a recent post I described a method of generating the simplest primitive Pythagorean triple (a,b,c) where one of the angles of the triangle with sides a, b and c is θ° to within some (small) error bound Δθ°.

One of the steps was, given the cosine C of the angle [from step (2)]:

(3) Calculate the Farey ratio approximant …

…

Now, that function

seemed semi-familiar, resembling functions that occur in trigonometric or hyperbolic identities.

An example is:

A little further investigation, and reading around, including the Wikipedia articles on trigonometric identities, and in particular on those of the tangent half-angle, revealed that the *Farey ratio approximant* does in fact correspond directly to a simple trigonometric function of the angle:

The slightly simplified method follows.