Here is another typical example. This proves that fights with me. Because of the method of solution.

http://mathoverflow.net/questions/31118/integer-polynomials-taking-square-values

Ask for exactly the formula for solving the equation.  I wrote a formula linking this equation with equation Pell.  I wrote the formula in General. You can substitute any of the coefficients. These formulas look quite simple.


For these equations we use the standard approach. 

For a private quadratic form:  $$Y^2=aX^2+bX+1$$    

Using solutions of Pell's equation:  $$p^2-as^2=1$$    

Solutions can be expressed through them is quite simple.    

$$Y=p^2+bps+as^2$$    $$X=2ps+bs^2$$    

$p,s$ - these numbers can have any sign.  Finding solutions of equations Pell - standard procedure.

Most interesting is that all the examples which lead shows the relationship of this equation with Equation Pell.  What is the point to write thousands of sequences ? If there's one formula that describes them all.

Bring as an example the numbers, and it is good. When I write a formula - it require to remove!