A246903 Irregular triangular array: row n lists the numbers D, each being the discriminant of the minimal polynomial of a quadratic irrational represented by a continued fraction with period an n-tuple of 1s and 2s.

5, 8, 5, 8, 12, 5, 8, 40, 85, 5, 8, 12, 96, 221, 480, 5, 8, 145, 260, 533, 1160, 1300, 2813, 5, 8, 12, 40, 85, 672, 1365, 1517, 1680, 3132, 3360, 7565, 16380, 5, 8, 901, 1768, 3725, 3973, 4625, 4901, 7400, 8104, 8468, 8840, 16133, 18229, 39208, 40004, 44104

Offset

1,1

Links

Table of n, a(n) for n=1..53.

Example

First 5 rows:
5 ... 8
5 ... 8 ... 12
5 ... 8 ... 40 .. 85
5 ... 8 ... 12 .. 96 .. 221 . 480
5 ... 8 ... 145 . 260 . 533. 1160 . 1300 . 2813
The following list shows for n = 3 the purely periodic continued fractions (with period an n-tuple of 1s and 2s), each followed by the number r it represents, the minimal polynomial a*x^2 + b*x + c of r, and the discriminant, D = b^2 - 4*a*c.
[(1,1,1)] = (1+sqrt(5))/2, -1 - x + x^2, D = 5
[(1,1,2)] = sqrt(10)/4, -5 + 2 x^2, D = 40
[(1,2,1)] = (2 + sqrt(10)/3, -2 - 4 x + 3 x^2, D = 10
[(2,1,1)] = (1 + sqrt(85))/6, -7 - x + 3 x^2, D = 85
[(1,2,2)] = (1 + sqrt(10)/3, -3 - 2 x + 3 x^2, D = 10
[(2,1,2)] = (-1 + sqrt(85))/6, -7 + x + 3 x^2, D = 85
[(2,2,2)] = (5 + sqrt(85))/10, -3 - 5 x + 5 x^2, D = 85
[(2,2,2)] = sqrt(2), -2 + x^2, D = 8
The distinct values of D are 5, 8, 10, 85, as in row 3.

Mathematica

z = 7; u[n_] := u[n] = Table[MinimalPolynomial[Map[FromContinuedFraction[{1, #}] &, Tuples[{1, 2}, k]], x], {k, 1, n}]; d = Discriminant[u[z], x];
t = Table[Union[d[[n]]], {n, 1, z}]; TableForm[t] (* A246903 array *)
Flatten[t] (* A246903 sequence *)

Crossrefs

Cf. A246904, A246905.

Sequence in context: A100610 A101465 A010719 * A213022 A198732 A202348

Adjacent sequences: A246900 A246901 A246902 * A246904 A246905 A246906

Keyword

nonn,tabf,easy

Author

Clark Kimberling, Sep 06 2014

Status

approved