Curious Corridors of Numbers

Curious Corridors of Numbers Each row begins with the number 1 at the left. Skip counting by 2, 3, 4, 5, and so on gives the rest of each row. The diagram can be continued as far as desired. - idea from Math Puzzles & Games, by Michael Holt, pg 27

Adding the numbers in each L-shaped region forms a well known mathematical sequence. What pattern shows up? Another mathematical pattern is formed by the numbers along the diagonal. What pattern is this?