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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.

%I A246903 #27 Dec 06 2024 11:20:53
%S A246903 5,8,5,8,12,5,8,40,85,5,8,12,96,221,480,5,8,145,260,533,1160,1300,
%T A246903 2813,5,8,12,40,85,672,1365,1517,1680,3132,3360,7565,16380,5,8,901,
%U A246903 1768,3725,3973,4625,4901,7400,8104,8468,8840,16133,18229,39208,40004,44104,95485
%N 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.
%e A246903 First 5 rows:
%e A246903   5 ... 8
%e A246903   5 ... 8 ... 12
%e A246903   5 ... 8 ... 40 .. 85
%e A246903   5 ... 8 ... 12 .. 96 .. 221 . 480
%e A246903   5 ... 8 ... 145 . 260 . 533 . 1160 . 1300 . 2813
%e A246903 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.
%e A246903 [(1,1,1)] = (1+sqrt(5))/2, -1 - x + x^2, D = 5
%e A246903 [(1,1,2)] = sqrt(5/2), -5 + 2 x^2, D = 40
%e A246903 [(1,2,1)] = (2 + sqrt(10))/3, -2 - 4 x + 3 x^2, D = 40
%e A246903 [(2,1,1)] = (1 + sqrt(10))/3, -3 - 2 x + 3 x^2, D = 40
%e A246903 [(1,2,2)] = (1 + sqrt(85))/6, -7 - x + 3 x^2, D = 85
%e A246903 [(2,1,2)] = (-1 + sqrt(85))/6, -7 + x + 3 x^2, D = 85
%e A246903 [(2,2,1)] = (5 + sqrt(85))/10, -3 - 5 x + 5 x^2, D = 85
%e A246903 [(2,2,2)] = sqrt(2), -2 + x^2, D = 8
%e A246903 The distinct values of D are 5, 8, 10, 85, as in row 3.
%t A246903 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 A246903 t = Table[Union[d[[n]]], {n, 1, z}]; TableForm[t] (* A246903 array *)
%t A246903 Flatten[t] (* A246903 sequence *)
%Y A246903 Cf. A246904, A246905.
%K A246903 nonn,tabf,easy
%O A246903 1,1
%A A246903 _Clark Kimberling_, Sep 06 2014
%E A246903 Edited by _Clark Kimberling_, Dec 05 2024