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.
Original entry on oeis.org
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
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.
-
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 *)
A246903
Irregular triangular array: row n lists the numbers D, each being the discriminant of the minimal polynomial of a quadratic irrational represented by a continued fraction with period an n-tuple of 1s and 2s.
Original entry on oeis.org
5, 8, 5, 8, 12, 5, 8, 40, 85, 5, 8, 12, 96, 221, 480, 5, 8, 145, 260, 533, 1160, 1300, 2813, 5, 8, 12, 40, 85, 672, 1365, 1517, 1680, 3132, 3360, 7565, 16380, 5, 8, 901, 1768, 3725, 3973, 4625, 4901, 7400, 8104, 8468, 8840, 16133, 18229, 39208, 40004, 44104, 95485
Offset: 1
First 5 rows:
5 ... 8
5 ... 8 ... 12
5 ... 8 ... 40 .. 85
5 ... 8 ... 12 .. 96 .. 221 . 480
5 ... 8 ... 145 . 260 . 533 . 1160 . 1300 . 2813
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.
[(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 5, 8, 10, 85, as in row 3.
-
z = 7; u[n_] := u[n] = Table[MinimalPolynomial[Map[FromContinuedFraction[{1, #}] &, Tuples[{1, 2}, k]], x], {k, 1, n}]; d = Discriminant[u[z], x];
t = Table[Union[d[[n]]], {n, 1, z}]; TableForm[t] (* A246903 array *)
Flatten[t] (* A246903 sequence *)
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.
Original entry on oeis.org
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
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.)
-
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 *)
Showing 1-3 of 3 results.
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