I have deleted my previous answer and instead posting this one, which is much more interesting. I investigated the sequence $Y_k=X_{3k}$, which has far more communal planes, and thus more useful to build a random generator. Of course, choosing large values for $b_1,b_2$ will further drastically improve the generator by adding a lot more planes. Here $b_1=5, b_2=3$. I suggest choosing values larger than (say) $2^{30}$ for $b_1,b_2$. Also $X_1=\frac{\sqrt{2}}{2}$.  

I found 197 communal planes (there might be a couple more) and they all have an equation of the form

 $$Y_k+0.576 \cdot Y_{k+1}-0.008\cdot Y_{k+2} =c.$$

The intercept $c$ ranges from $0$ to $1.568$ by equal increments of $0.008$. Each plane (identified by $c$) contains a different proportion of triplets $(Y_k,Y_{k+1},Y_{k+2})$. The empirical distribution for these proportions is featured in the histogram below, where the X-axis represents $c$, and the Y-axis the proportion of triplets lying in plan $c$.

[![enter image description here][1]][1]

Of course it is easy by looking at this chart to guess what the exact theoretical distribution is. To identify these planes, I used the program below.

    # Compute equations of planes containing 3 random vectors
    #   P(k) = (x[k], x[k+1], x[k+2])
    #   P(l) = (x[l], x[l+1], x[l+2])
    #   P(m) = (x[m], x[m+1], x[m+2])
    # (k, l, m) are randomly selected (M triplets)
    #
    # Equation of planes is x + s*y + t*z = intercept
    # For each (k,l,m) output the coefficients s, t, intercept 
    #
    # Goal: Find communal planes absorbing many (P(k), P(l), P(m))
    # Once the planes are computed, sort them by s, t, intercept
    
    $n=100000;
    
    $b1=5; 
    $b2=3;
    
    # xx[] is the original sequence
    
    $xx[0]=0.5;
    $xx[1]=sqrt(2)/2;
    
    for ($k=2; $k<$n; $k++) {
      $xx[$k]=$b2*$xx[$k-1]+$b1*$xx[$k-2]-int($b2*$xx[$k-1]+$b1*$xx[$k-2]); 
      if ($xx[$k]<0) { $xx[$k]=1+$xx[$k]; }
    }
    
    # we actually use 1 out of 3 consecutive terms from original sequence xx[]
    # to see if it the new sequence x[] also has a small number of communal planes 
    
    for ($k=0; $k< $n/3; $k++) {
      $x[$k]=$xx[3*$k];
    }
    
    $M=10000; # must be < n/3
    open(OUT,">coplanes2.txt");
    
    for ($iter=0; $iter<$M; $iter++) {
    
           $k=int($M*rand()); 
           $l=int($M*rand());
           $m=int($M*rand());
    
           # in case k=l or k=m or l=m, an ERROR message is reported
    
           $a=$x[$k]; $b=$x[$k+1]; $c=$x[$k+2];
           $d=$x[$l]; $e=$x[$l+1]; $f=$x[$l+2];
           $p=$x[$m]; $q=$x[$m+1]; $r=$x[$m+2];
           $u=($e-$b)*($r-$c)-($f-$c)*($q-$b);
           $v=-($d-$a)*($r-$c)+($f-$c)*($p-$a);
           $w=($d-$a)*($q-$b)-($e-$b)*($p-$a);
    
           if ($u != 0) {
             $s=$v/$u;
             $t=$w/$u;
             $intercept=($u*$a + $v*$b + $w*$c)/$u;
    
             print OUT "$k\t$l\t$m\t";
             print OUT "$s\t$t\t$intercept\n";
    
           } else {
             print OUT "$k\t$l\t$m\tERROR (u=0)\n";
           }
    }  
    close(OUT); 

   


  [1]: https://i.sstatic.net/h46bI.png