A246922 Irregular triangular array: every periodic simple continued fraction CF represents a quadratic irrational (c + f*sqrt(d))/b, where b,c,f,d are integers and d is squarefree. Row n of this array shows the distinct values of d as CF ranges through the periodic continued fractions having period an n-tuple of 1s and 3s.

5, 13, 5, 13, 21, 5, 13, 17, 65, 5, 13, 21, 29, 165, 2805, 5, 13, 61, 317, 445, 1853, 5933, 30629, 2, 5, 7, 13, 15, 17, 21, 34, 35, 65, 66, 145, 5402, 5, 13, 3029, 10205, 11029, 12773, 28157, 34973, 42853, 47965, 53365, 136165, 184045, 187493, 219965, 724205

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 .. 29 ... 165 .. 2805
  5 ... 13 ... 61 .. 317 .. 445 .. 1853 .. 5933 .. 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, and the squarefree factor, d, of D.
[(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.  (Here, d = D for all entries, but higher numbered rows, this d < D for some entried.)
		

Crossrefs

Cf. A246905, A246921.

Programs

Mathematica

z = 8; t[n_] := t[n] = Map[FromContinuedFraction[{1, #}] &, Tuples[{1, 3}, n]]; u[n_] := u[n] = Table[MinimalPolynomial[t[k], x], {k, 1, n}]; d = Discriminant[u[z], x];
v[n_] := Table[{p, m} = Transpose[FactorInteger[k]]; Times @@ (p^Mod[m, 2]), {k, d[[n]]}]; w = Table[Union[Table[v[n], {n, 1, z}][[n]]], {n, 1, z}]; TableForm[w] (* A246922 array *)
Flatten[w]          (* A246922 sequence *)

Extensions

Edited by Clark Kimberling, Dec 05 2024