A246903 - OEIS

%I #27 Dec 06 2024 11:20:53

%S 5,8,5,8,12,5,8,40,85,5,8,12,96,221,480,5,8,145,260,533,1160,1300,

%T 2813,5,8,12,40,85,672,1365,1517,1680,3132,3360,7565,16380,5,8,901,

%U 1768,3725,3973,4625,4901,7400,8104,8468,8840,16133,18229,39208,40004,44104,95485

%N 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 First 5 rows:

%e   5 ... 8

%e   5 ... 8 ... 12

%e   5 ... 8 ... 40 .. 85

%e   5 ... 8 ... 12 .. 96 .. 221 . 480

%e   5 ... 8 ... 145 . 260 . 533 . 1160 . 1300 . 2813

%e 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 [(1,1,1)] = (1+sqrt(5))/2, -1 - x + x^2, D = 5

%e [(1,1,2)] = sqrt(5/2), -5 + 2 x^2, D = 40

%e [(1,2,1)] = (2 + sqrt(10))/3, -2 - 4 x + 3 x^2, D = 40

%e [(2,1,1)] = (1 + sqrt(10))/3, -3 - 2 x + 3 x^2, D = 40

%e [(1,2,2)] = (1 + sqrt(85))/6, -7 - x + 3 x^2, D = 85

%e [(2,1,2)] = (-1 + sqrt(85))/6, -7 + x + 3 x^2, D = 85

%e [(2,2,1)] = (5 + sqrt(85))/10, -3 - 5 x + 5 x^2, D = 85

%e [(2,2,2)] = sqrt(2), -2 + x^2, D = 8

%e The distinct values of D are 5, 8, 10, 85, as in row 3.

%t 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 t = Table[Union[d[[n]]], {n, 1, z}]; TableForm[t] (* A246903 array *)

%t Flatten[t] (* A246903 sequence *)

%Y Cf. A246904, A246905.

%K nonn,tabf,easy

%O 1,1

%A _Clark Kimberling_, Sep 06 2014

%E Edited by _Clark Kimberling_, Dec 05 2024