Integers $d$ for which the Negative Pell equation is soluble for both $d$ and $2d$? Let $\text{NPE}_d$ denote the negative Pell equation:
$$ x^2-dy^2=-1$$
Where $d$ is a given positive nonsquare integer and integer solutions are sought for x and y.

we know that (in this paper):


  Theorem : The equation $\text{NPE}_d$ has integer solutions if and only if there exist two integers $a(d)=a$ and $b(d)=b$ such that $d=a^2+b^2$ and there exists a Pythagorean triplet $(A,B,C)$
  such that $|aA-bB|=1$ and in this case $(Aa+Bb,C)$ is a solution.


Obviously if $\text{NPE}_d$ has integer solutions then $d$ cannot be divisible by any prime $p$ such that $p=3\mod 4$.


  My question: Is there any characterization for the integers $d$ for which $\text{NPE}_d$ and $\text{NPE}_{2d}$ have both integer solutions.


I used the characterization above, but I can't link the couple $(a(d),b(d))$ 
to $(a(2d),b(2d))$ because the theorem doesn't give us much information

The sequence  of the elements $d$ for which $\text{NPE}_d$  is soluble is OEIS A031396.

Thank you for your help.
It is necessary to use the formula.    http://www.artofproblemsolving.com/blog/101140

$$x^2-dy^2=-1$$

$$z^2-2dg^2=-1$$

It will give us a series of solutions using the Pell equation:  $$p^2-2s^2=k$$

Let us use the fact that the following solution can be found knowing the previous, formula.

$$p_2=3p_1+4s_1$$

$$s_2=2p_1+3s_1$$

Using certain series of solutions you can find ratio formula.

$p^2-2s^2=1$     $;$     $(p_1;s_1) - (3;2);$  $d=s^2+1$

$p^2-2s^2=-1$   $;$   $(p_1;s_1) - (1;1) ;$   $d=s^2+4$

$p^2-2s^2=-7$    $;$   $(p_1;s_1)  - (1;2) ;$    $d=s^2+1$

$p^2-2s^2=7$    $;$   $(p_1;s_1) - (3;1) ;$   $d=s^2+4$

$p^2-2s^2=17$   $;$   $(p_1 ; s_1)  - (5;2) ;$  $d=s^2+9$

While it is difficult to say. Limited to any solution of these series or not.  Clearly what is involved in this equation Pell.
It is necessary to use the formula.    http://www.artofproblemsolving.com/blog/101140

$$x^2-dy^2=-1$$

$$z^2-2dg^2=-1$$

It will give us a series of solutions using the Pell equation:  $$p^2-2s^2=k$$

Let us use the fact that the following solution can be found knowing the previous, formula.

$$p_2=3p_1+4s_1$$

$$s_2=2p_1+3s_1$$

Using certain series of solutions you can find ratio formula.

$p^2-2s^2=1$     $;$     $(p_1;s_1) - (3;2);$  $d=s^2+1$

$p^2-2s^2=-1$   $;$   $(p_1;s_1) - (1;1) ;$   $d=s^2+4$

$p^2-2s^2=-7$    $;$   $(p_1;s_1)  - (1;2) ;$    $d=s^2+1$

$p^2-2s^2=7$    $;$   $(p_1;s_1) - (3;1) ;$   $d=s^2+4$

$p^2-2s^2=17$   $;$   $(p_1 ; s_1)  - (5;2) ;$  $d=s^2+9$

While it is difficult to say. Limited to any solution of these series or not.  Clearly what is involved in this equation Pell.