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What do these things all have in common? Anime and Manga of course!
So does anyone have any knowledge to spread on these simularities?
Especially with the 26/13 ep thing coz im a big anime/manga fan i wanna know

Less than 13= OAV
13 episodes = TV series
26 episodes = One season
52 episodes = Two seasons

Basically, the # of ep. in an anime series are usually in multiples of 13.
Sometimes shows will have 24, 14, 17, etc. But it's usually the former.

As for cats and paper fans...

The "paper-fan-attack" is the most common style of slap-stick comedy in Japan. So it's only natural to see it in comedy anime

A kawaii neko (cute cat) is a popular icon in Japanese Pop-culture. So in an anime with lots of cute things, a cat is bound to pop up somewhere!

I don't know about Japanese comedy, but another reason fans show up a lot because they were very important in the samurai tradition; people did, in fact, fight with fans (called tessen). In addition to being important cerimonially, samurai could carry them in places where etiquite would prevent them from carrying their sword (such as when meeting with their superiors), and thus avoid being unarmed..

Just to elaborate more on that. 52 weeks in a year thats why that number is picked. Just wasn't picked out of the air.

Originally posted by Lumenesca
As for cats and paper fans...

The "paper-fan-attack" is the most common style of slap-stick comedy in Japan. So it's only natural to see it in comedy anime

A kawaii neko (cute cat) is a popular icon in Japanese Pop-culture. So in an anime with lots of cute things, a cat is bound to pop up somewhere!
Naruto's Temari with a paper fan.

A Neko is in practically every anime or manga. Takuto (fullmmoon O Sagashite), Kyo (FB), the stuffed animal from Card Captors.