When you upload an image, it’s saved, and a copy is created for every image size, resized to the appropriate dimensions.
When you specify a set of dimensions manually, e.g.
wp_get_attachment_image_src($id, array(57, 57))
the 57×57 dimensions are compared to the image sizes, and the nearest image size is chosen and that image is returned. So for that to work you’d need to define image sizes for each stage.
From your question I believe you want WordPress to generate these when they’re first requested though, not on upload.
If so, this plugin will do the trick:
http://wordpress.org/extend/plugins/wp-thumb/
It replaces the image handling internally. With that installed image handling should work a lot closer to what you’re expecting, e.g. you would get a 57px by 57px image not the nearest size