Generating Pythagorean Triples

Suppose that m and n are two positive integers, with m < n. Then n² - m², 2mn, and n² + m² is a Pythagorean triple.

It's easy to check algebraically that the sum of the squares of the first two is the same as the square of the last one. Is it the case that every triple can be generated in this manner?

Here are the first few triples for m and n between 1 and 10. Notice any patterns?

  
   | m= 1          2           3           4            5           6              7            8            9
---+------------------------------------------------------------------------------------------------------------------
n= |
 2 | [3,4,5]
   | 
 3 | [8,6,10]    [5,12,13]
   | 
 4 | [15,8,17]   [12,16,20]  [7,24,25]
   | 
 5 | [24,10,26]  [21,20,29]  [16,30,34]  [9,40,41]
   | 
 6 | [35,12,37]  [32,24,40]  [27,36,45]  [20,48,52]  [11,60,61]
   | 
 7 | [48,14,50]  [45,28,53]  [40,42,58]  [33,56,65]  [24,70,74]   [13,84,85]
   | 
 8 | [63,16,65]  [60,32,68]  [55,48,73]  [48,64,80]  [39,80,89]   [28,96,100]  [15,112,113]
   | 
 9 | [80,18,82]  [77,36,85]  [72,54,90]  [65,72,97]  [56,90,106]  [45,108,117] [32,126,130] [17,144,145]
   | 
10 | [99,20,101] [96,40,104] [91,60,109] [84,80,116] [75,100,125] [64,120,136] [51,140,149] [36,160,164] [19,180,181]