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, 95485

Offset: 1

Author

Clark Kimberling, Sep 06 2014

Keywords

• nonn
• tabf
• easy

Examples

			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(5/2), -5 + 2 x^2, D = 40
[(1,2,1)] = (2 + sqrt(10))/3, -2 - 4 x + 3 x^2, D = 40
[(2,1,1)] = (1 + sqrt(10))/3, -3 - 2 x + 3 x^2, D = 40
[(1,2,2)] = (1 + sqrt(85))/6, -7 - x + 3 x^2, D = 85
[(2,1,2)] = (-1 + sqrt(85))/6, -7 + x + 3 x^2, D = 85
[(2,2,1)] = (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.
		

Crossrefs

Cf. A246904, A246905.

Programs

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 *)

Extensions

Edited by Clark Kimberling, Dec 05 2024