How to write this function in a nicer way?

Hi all,

Here is a counting formula that has been derived using some recursion formulas but I think it should have a much more elegant form. Any help writing it in " a nicer form " would be really appreciated.

Here is the formula :

$P(n, m) = \lfloor \dfrac{m}{2} \rfloor (2 \lfloor \dfrac{n}{3} \rfloor + \dddot{n} ) + \ddot{m} { \{ \dfrac{n}{3} } \}$

where

$\dddot{n}$ is the the remainder of n mod 3,
$\ddot{m}$ is the the remainder of m mod 2,
$\{ k \}$ is the integer part of the rational number k.

I expect this function (if derived correctly) is symmetric s.t. P(n,m) = P(m,n) but its no obvious in this current form.

Thanks and enjoy...

EDIT : I found a mistake in the derivation, please ignore :)

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$