

A246921


Irregular triangular array: row n gives numbers D, each being the discriminant of the minimal polynomial of a quadratic irrational represented by a continued fraction with period an ntuple of 1s and 3s.


2



5, 13, 5, 13, 21, 5, 13, 17, 65, 5, 13, 21, 165, 725, 2805, 5, 13, 445, 1525, 1853, 5933, 7925, 30629, 5, 13, 17, 21, 65, 136, 288, 960, 1260, 4224, 16128, 21608, 83520, 5, 13, 3029, 10205, 11029, 12773, 28157, 34973, 42853, 47965, 53365, 136165, 184045
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OFFSET

1,1


LINKS



EXAMPLE

First 5 rows:
5 ... 13
5 ... 13 ... 21
5 ... 13 ... 17 .. 65
5 ... 13 ... 21 .. 165 .. 725 .. 2805
5 ... 13 ... 445 . 1525 . 1853 . 5933 . 7925 . 30629
The following list shows for n = 3 the purely periodic continued fractions (with period an ntuple 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
[(1,1,3)] = (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[65])/4, 8 + x + 2 x^2, D = 65
[(1,3,3)] = (1 + sqrt(17)/4, 2  x + 2 x^2, D = 17
[(3,1,3)] = (3 + sqrt(65))/4, 7 + 3 x + 2 x^2, D = 65
[(3,3,3)] = (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.


MATHEMATICA

z = 7; u[n_] := u[n] = Table[MinimalPolynomial[Map[FromContinuedFraction[{1, #}] &, Tuples[{1, 3}, k]], x], {k, 1, n}]; d = Discriminant[u[z], x];
t = Table[Union[d[[n]]], {n, 1, z}]; TableForm[t] (* A246921 array *)


CROSSREFS



KEYWORD

nonn,tabf,easy


AUTHOR



STATUS

approved



