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 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. 2

%I #7 Sep 13 2014 14:55:36

%S 5,13,5,13,21,5,13,17,65,5,13,21,29,165,2805,5,13,61,317,445,1853,

%T 5933,30629,2,5,7,13,15,17,21,34,35,65,66,145,5402,5,13,3029,10205,

%U 11029,12773,28157,34973,42853,47965,53365,136165,184045,187493,219965,724205

%N 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.

%e First 5 rows:

%e 5 ... 13

%e 5 ... 13 ... 21

%e 5 ... 13 ... 17 .. 65

%e 5 ... 13 ... 21 .. 29 ... 165 .. 2805

%e 5 ... 13 ... 61 .. 317 .. 445 .. 1853 .. 5933 .. 30629

%e 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.

%e [(1,1,1)] = (1+sqrt(5))/2, -1 - x + x^2, D = 5 = d

%e [(1,1,3)] = (-1 + sqrt(17)/2, -4 + x + x^2, D = 17 = d

%e [(1,3,1)] = (3 + sqrt(17)/4, -1 - 3 x + 2 x^2, D = 17 = d

%e [(3,1,1)] = (-1 + Sqrt[65])/4, -8 + x + 2 x^2, D = 65 = d

%e [(1,3,3)] = (1 + sqrt(17)/4, -2 - x + 2 x^2, D = 17 = d

%e [(3,1,3)] = (-3 + sqrt(65))/4, -7 + 3 x + 2 x^2, D = 65 = d

%e [(3,3,3)] = (5 + sqrt(65))/10, -2 - 5 x + 5 x^2, D = 65 = d

%e [(3,3,3)] = (-1 + sqrt(13))/2, -3 + x + x^2, D = 13 = d

%e 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.)

%t 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];

%t 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 *)

%t Flatten[w] (* A246922 sequence *)

%Y Cf. A246905, A246921.

%K nonn,tabf,easy

%O 1,1

%A _Clark Kimberling_, Sep 07 2014

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Last modified April 18 02:22 EDT 2024. Contains 371767 sequences. (Running on oeis4.)