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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.
3
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
OFFSET
1,1
EXAMPLE
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(10)/4, -5 + 2 x^2, D = 40
[(1,2,1)] = (2 + sqrt(10)/3, -2 - 4 x + 3 x^2, D = 10
[(2,1,1)] = (1 + sqrt(85))/6, -7 - x + 3 x^2, D = 85
[(1,2,2)] = (1 + sqrt(10)/3, -3 - 2 x + 3 x^2, D = 10
[(2,1,2)] = (-1 + sqrt(85))/6, -7 + x + 3 x^2, D = 85
[(2,2,2)] = (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.
MATHEMATICA
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 *)
CROSSREFS
Sequence in context: A100610 A101465 A010719 * A213022 A198732 A202348
KEYWORD
nonn,tabf,easy
AUTHOR
Clark Kimberling, Sep 06 2014
STATUS
approved