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 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 n-tuple 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 LINKS Table of n, a(n) for n=1..49. 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 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 [(1,1,3)] = (-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 *) Flatten[t] (* A246921 sequence *) CROSSREFS Cf. A246904, A246922. Sequence in context: A297903 A298497 A246922 * A170864 A293958 A089619 Adjacent sequences: A246918 A246919 A246920 * A246922 A246923 A246924 KEYWORD nonn,tabf,easy AUTHOR Clark Kimberling, Sep 07 2014 STATUS approved

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Last modified February 25 06:24 EST 2024. Contains 370310 sequences. (Running on oeis4.)