CF1393D Rarity and New Dress

Carousel Boutique is busy again! Rarity has decided to visit the pony ball and she surely needs a new dress, because going out in the same dress several times is a sign of bad manners. First of all, she needs a dress pattern, which she is going to cut out from the rectangular piece of the multicolored fabric. The piece of the multicolored fabric consists of $ n \times m $ separate square scraps. Since Rarity likes dresses in style, a dress pattern must only include scraps sharing the same color. A dress pattern must be the square, and since Rarity is fond of rhombuses, the sides of a pattern must form a $ 45^{\circ} $ angle with sides of a piece of fabric (that way it will be resembling the traditional picture of a rhombus). Examples of proper dress patterns:  Examples of improper dress patterns:  The first one consists of multi-colored scraps, the second one goes beyond the bounds of the piece of fabric, the third one is not a square with sides forming a $ 45^{\circ} $ angle with sides of the piece of fabric. Rarity wonders how many ways to cut out a dress pattern that satisfies all the conditions that do exist. Please help her and satisfy her curiosity so she can continue working on her new masterpiece!

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In the first example, all the dress patterns of size $ 1 $ and one of size $ 2 $ are satisfactory. In the second example, only the dress patterns of size $ 1 $ are satisfactory.