If you square each number, subtract one square from the square greater than it, then square root this number, you can find out what Pythagorean Triples there are. If the result is a whole number, the two numbers and the square rooted number make up a Pythagorean Triple. For example, 24 squared is 576, and 25 squared is 625. 625-576=49. 49 is a square number (7 squared) so these make up a Pythagorean Triple. 7 squared + 24 squared = 25 squared.
The first square, when the hypotenuse is one greater than the other side, is always odd, increasing by two each time. For example:
9 squared + 41 squared = 42 squared
11 squared + 60 squared = 61 squared
13 squared + 84 squared = 85 squared
If, instead of subtracting the square from the square one greater than it, you subtract it from the square two greater, you can work out the Pythagorean Triples similar to Alison's. For example, 15 squared is 225 and 17 squared is 289. 289 - 225 = 64. 64 is a square number (8 squared) so this makes up a Pythagorean Triple. 8 squared + 15 squared = 17 squared
The first square number is always even and increases by two each time. For example:
10 squared + 24 squared = 26 squared
12 squared + 35 squared = 37 squared
14 squared + 48 squared = 50 squared
If the hypotenuse is three longer than a side, the shortest side increases by 6 each time (starting on 3) e.g. 3, 15, 21, 27 etc. If the hypotenuse is four units linger, the shortest side increases by 4 each time e.g. 4, 8, 12, 16 etc.
The spreadsheet contains the four different unit sizes, with numbers from around 15 - 1000. The shortest sides are highlighted yellow.