A246904 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 2s.
2, 5, 2, 3, 5, 2, 5, 10, 85, 2, 3, 5, 6, 30, 221, 2, 5, 13, 65, 145, 290, 533, 2813, 2, 3, 5, 10, 42, 85, 87, 105, 210, 455, 1365, 1517, 7565, 2, 5, 29, 58, 74, 149, 185, 442, 565, 901, 2026, 2117, 2210, 3973, 10001, 11026, 16133, 18229, 2, 3, 5, 6, 26, 30
Offset: 1
Examples
First 5 rows: 2 ... 5 2 ... 3 ... 5 2 ... 5 ... 10 .. 85 2 ... 3 ... 5 ... 6 ... 30 ... 221 2 ... 5 ... 13 .. 65 .. 145 .. 290 .. 533 .. 2813 The following list shows for n = 3 the 2^n purely periodic continued fractions, each followed by the number r it represents, the minimal polynomial a*x^2 + b*x + c of r, 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 [(1,1,2)] = 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 2, 5, 10, 85, as in row 3.
Programs
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Mathematica
z = 6; t[n_] := t[n] = Map[FromContinuedFraction[{1, #}] &, Tuples[{1, 2}, 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] (* A246904 array *) Flatten[w] (* A246904 sequence *)
Extensions
Edited by Clark Kimberling, Dec 05 2024
Comments