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

1,1

Links

Table of n, a(n) for n=1..52.

Example

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.)

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 *)

Crossrefs

Cf. A246905, A246921.

Sequence in context: A057691 A297903 A298497 * A246921 A170864 A293958

Adjacent sequences: A246919 A246920 A246921 * A246923 A246924 A246925

Keyword

nonn,tabf,easy

Author

Clark Kimberling, Sep 07 2014

Extensions

Edited by Clark Kimberling, Dec 05 2024

Status

approved