A246905 Number of numbers in row n of A246904.

2, 3, 4, 6, 8, 13, 18, 30, 46, 78, 125, 220, 374

Offset

1,1

Comments

Each periodic continued fraction with period an n-tuple of 1s and 2s represents a number r in a quadratic number field, Q(x), where x is a squarefree positive integer, as in A246904, and a(n) = number of distinct numbers x.

Links

Table of n, a(n) for n=1..13.

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}]]

Crossrefs

Cf. A246903, A246904.

Sequence in context: A039880 A240452 A263359 * A000029 A155051 A018137

Adjacent sequences: A246902 A246903 A246904 * A246906 A246907 A246908

Keyword

nonn,more

Author

Clark Kimberling, Sep 06 2014

Status

approved