We have:

because

In an earlier post, I showed the finite version of this result:

This might be seen to hold as both sums are

Mathematics — Algorithms — Version Control

We have:

because

In an earlier post, I showed the finite version of this result:

This might be seen to hold as both sums are

Whilst doodling with the Fibonacci sequence

n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | … |
---|---|---|---|---|---|---|---|---|---|---|---|---|

F_{n} |
0 | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 | 55 | … |

I found some interesting formulae:

The first of these is not new, but I did not find the other two on the web.

There’s more information about the Fibonacci sequence and the binomial coefficients on Wikipedia.

where ω is a primitive cube root of 1 given by

are often represented in the form

(noting that )

and perhaps abbreviated by pair notation such as

(where the asterisk is to distinguish this notation from that introduced below)

so that

Here is a more intuitive representation that is simpler to manipulate and reason about.

To colour the current tab green, and any tab of an edited file red, add the following to your theme file.

If you have problems remembering which is which with **sinh** and **cosh**, or their graphs, or which way around their definitions are:

then it might help if you notice that the graph of **sinh** is somewhat *S*-shaped (though backwards), and **cosh** is *C*-shaped (though open upwards).

[Well, unless you’re Russian, where the Cyrillic letter *Es* (‘С’) corresponds to our ‘S’; that might throw a graphemic spanner in the works.]

Here is a summary of rules of derivation (sometimes referred to as differentiation).

I wrote this post as the notation used on other sites is often ambiguous (particularly with juxtapositions that could denote either a product or a function application).

I previously wrote some brief notes on some derivations of the derivation formulae.

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.

In my previous post on Fibonacci-style sequences, I gave the formulae for the closed form

given the recurrence values where

but I did not show that the formula is valid. I do that here.

The Fibonacci sequence which begins is defined by

(the starting values vary by author).

This is a special case of a more general sequence given by

There are at least three ways to determine such a sequence. I explore those here.

Serious, technical books I recommend.

These are listed in no particular order

**Genetics**

The Continuity of Life

*Daniel J. Fairbanks, W. Ralph Andersen*

**Lectures on Physics**

*Richard Feynman*

**Concrete Mathematics**

*Ronald L. Graham, Donald E. Knuth and Oren Patashnik*