A246921 Irregular triangular array: row n gives 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 3s.

5, 13, 5, 13, 21, 5, 13, 17, 65, 5, 13, 21, 165, 725, 2805, 5, 13, 445, 1525, 1853, 5933, 7925, 30629, 5, 13, 17, 21, 65, 136, 288, 960, 1260, 4224, 16128, 21608, 83520, 5, 13, 3029, 10205, 11029, 12773, 28157, 34973, 42853, 47965, 53365, 136165, 184045

Offset: 1

Author

Clark Kimberling, Sep 07 2014

Keywords

• nonn
• tabf
• easy

Examples

			First 5 rows:
  5 ... 13
  5 ... 13 ... 21
  5 ... 13 ... 17 .. 65
  5 ... 13 ... 21 .. 165 .. 725 .. 2805
  5 ... 13 ... 445 . 1525 . 1853 . 5933 . 7925 . 30629
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
[(1,1,3)] = (-1 + sqrt(17))/2, -4 + x + x^2, D = 17
[(1,3,1)] = (3 + sqrt(17))/4, -1 - 3 x + 2 x^2, D = 17
[(3,1,1)] = (1 + sqrt(17))/4, -2 - x + 2 x^2, D = 17
[(1,3,3)] = (-1 + sqrt(65))/4, -8 + x + 2 x^2, D = 65
[(3,1,3)] = (-3 + sqrt(65))/4, -7 + 3 x + 2 x^2, D = 65
[(3,3,1)] = (5 + sqrt(65))/10, -2 - 5 x + 5 x^2, D = 65
[(3,3,3)] = (-1 + sqrt(13))/2, -3 + x + x^2, D = 13
The distinct values of D are 5, 13, 17, 65, as in row 3.
		

Crossrefs

Cf. A246904, A246922.

Programs

Mathematica

z = 7; u[n_] := u[n] = Table[MinimalPolynomial[Map[FromContinuedFraction[{1, #}] &, Tuples[{1, 3}, k]], x], {k, 1, n}]; d = Discriminant[u[z], x];
t = Table[Union[d[[n]]], {n, 1, z}]; TableForm[t] (* A246921 array *)
Flatten[t] (* A246921 sequence *)

Extensions

Edited by Clark Kimberling, Dec 05 2024