To calculate the number of Pythagorean triples with no component larger than 300, we can use the formula: (1/2)*11*12 = 66. This formula is derived from the pattern observed on the website: http://www2.math.uic.edu/~fields/puzzle/triples.html. The largest side of these triples is 11^2 + 12^2 = 265. There are 11 integer pairs that satisfy this condition.
Next, to find the sum of the hypotenuses from these triples, we can calculate (1^2 + 2^2) + (2^2 + 3^2) + (3^2 + 4^2) + ... ((11^2 + 12^2). This simplifies to 1 + 2(4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 + 121) + 144 = 1155.
Finally, we add the results from the first two steps, 66 + 1155 = 1221.
Therefore, the answer is 1221.
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