Archive for November, 2013

Fibonacci Formulae

Thursday, 21 November 2013

Whilst doodling with the Fibonacci sequence

n 0 1 2 3 4 5 6 7 8 9 10
Fn 0 1 1 2 3 5 8 13 21 34 55

I found some interesting formulae:

  • F_n = \displaystyle\sum_{k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor} \binom{n-1-k}{k}
  • F_{2m} = - \dfrac{1}{2} \displaystyle\sum_{k=0}^{2m-1} (-1)^k \binom{2m}{k} F_k
  • \displaystyle\sum_{k=0}^{2m} (-1)^k \binom{2m+1}{k} F_k = 0

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.

 

An Intuitive Representation of the Eisenstein Integers

Wednesday, 6 November 2013

The Eisenstein integers

{\mathbb Z} [ \omega ]

where ω is a primitive cube root of 1 given by

\omega = \dfrac{-1 + {\mathrm {i}} \sqrt{3}}{2} = \sqrt[3]{1}

are often represented in the form

\{ a + b \cdot \omega \;|\; a, b \in {\mathbb Z}\}
(noting that \omega^2 = - (1 + \omega))

and perhaps abbreviated by pair notation such as

\{ [ a ; b ]^{*} \;|\; a, b \in {\mathbb Z}\}

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

so that

[ a ; b ]^{*} + [ c ; d ]^{*} = [a+b \;;\; c+d]^{*}

[ a ; b ]^{*} \times [ c ; d ]^{*} = [(a c - b d) \;;\; (b c + a d - b d) ]^{*}

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

(more…)