There's nothing built into core but it's easy to make something like this yourself. Here's an example that would give posts classes post-count-1, post-count-2, post-count-3, post-count-1, ....2, ....3, ...1, etc.:
\\ start the counter at 1
$counter = 1;
\\ start the loop
while->have_posts() : the_post();
\\ figure out our count
$count = 'post-count-' . $counter % 3;
<article <?php post_class( $count ); ?>>
\\ some more stuff like the title and content
\\ advance the counter
\\ keep looping until the end
If you wanted plain English classes, you'd need an
switch statement to set
$count. This example also uses
post_class which gives you some other useful classes.
I should add that if you don't need extensive browser support (:nth-child() CSS selector is really the right solution if you need the classes purely for styling. jQuery also supports this selector and you can use it to add the classes after page-load if you need better-than-the-CSS browser support.