Mathematical puzzles

Mathematical puzzles can perhaps always be akin to school homework or challenges to prove theorems in the style of the Elements of Euclid; still, they are more unusual than the school homework tasks the aforementioned represent. The following has not, as far as I know, been published anywhere else before except in SEMantics, the regional newsletter for the south east England of British Mensa.

Pythagorean triplets

Given any odd integer a, is it possible to derive a (larger, even) integer b such that, if = b + 1, a, b, c is a Pythagorean triplet; that is, such that a2 + b2 = c2 ?

Given any odd integer a, is it possible to derive (larger) integers b and c such that a, b, c is a Pythagorean triplet (that is, a2 + b2 = c2 ) but  b + 1 ?

Indeed, is it possible to derive a formula defining an entire series of such numbers b1, b2, b3, b4, ... from each of which c can be determined and is also still an integer?

Note: I do not have a formula as an answer, or indeed any answer (as to whether such a formula exists) for either of these questions. However, I suspect that the answer is that for many odd integers n there exists no Pythagorean triplet in which n appears.