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A246922
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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.
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2
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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
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OFFSET
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1,1
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LINKS
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EXAMPLE
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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 = d
[(1,1,3)] = (-1 + sqrt(17)/2, -4 + x + x^2, D = 17 = d
[(1,3,1)] = (3 + sqrt(17)/4, -1 - 3 x + 2 x^2, D = 17 = d
[(3,1,1)] = (-1 + Sqrt[65])/4, -8 + x + 2 x^2, D = 65 = d
[(1,3,3)] = (1 + sqrt(17)/4, -2 - x + 2 x^2, D = 17 = d
[(3,1,3)] = (-3 + sqrt(65))/4, -7 + 3 x + 2 x^2, D = 65 = d
[(3,3,3)] = (5 + sqrt(65))/10, -2 - 5 x + 5 x^2, D = 65 = d
[(3,3,3)] = (-1 + sqrt(13))/2, -3 + x + x^2, D = 13 = d
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.)
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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STATUS
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approved
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