
EXAMPLE

The following list shows for n = 3 the 2^n periodic continued fractions, each followed by the number r it represents, the minimal polynomial a*x^2 + b*x + c of r, the discriminant, D = b^2  4*a*c, and the squarefree factor, d, of D.
[(1,1,1)] = (1+sqrt(5))/2, 1  x + x^2, D = 5, d = 5
[(1,1,2)] = sqrt(10)/4, 5 + 2 x^2, D = 40, d = 10
[(1,2,1)] = (2 + sqrt(10)/3, 2  4 x + 3 x^2, D = 10, d = 10
[(2,1,1)] = (1 + sqrt(85))/6, 7  x + 3 x^2, D = 85, d = 85
[(1,2,2)] = (1 + sqrt(10)/3, 3  2 x + 3 x^2, D = 10, d = 10
[(2,1,2)] = (1 + sqrt(85))/6, 7 + x + 3 x^2, D = 85, d = 85
[(2,2,2)] = (5 + sqrt(85))/10, 3  5 x + 5 x^2, D = 85, d = 85
[(2,2,2)] = sqrt(2), 2 + x^2, D = 8, d = 2
The distinct values of d are 2, 5, 10, 85, so that A246905(3) = 4.


MATHEMATICA

z = 7; u[n_] := Table[MinimalPolynomial[Map[FromContinuedFraction[{1, #}] &, Tuples[{1, 2}, k]], x], {k, 1, n}]; Map[Length, Table[Union[Discriminant[u[z], x] [[n]]], {n, 1, z}]]
