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%I #7 Sep 13 2014 14:54:09
%S 2,5,2,3,5,2,5,10,85,2,3,5,6,30,221,2,5,13,65,145,290,533,2813,2,3,5,
%T 10,42,85,87,105,210,455,1365,1517,7565,2,5,29,58,74,149,185,442,565,
%U 901,2026,2117,2210,3973,10001,11026,16133,18229,2,3,5,6,26,30
%N Irregular triangular array: every periodic simple continued fraction CF represents a quadratic irrational (c + f*sqrt(d))/b, where b,c,f,d are integers and d is squarefree. Row n of this array shows the distinct values of d as CF ranges through the periodic continued fractions having period an n-tuple of 1s and 2s.
%C Is every squarefree positive integer in this array?
%e First 5 rows:
%e 2 ... 5
%e 2 ... 3 ... 5
%e 2 ... 5 ... 10 .. 85
%e 2 ... 3 ... 5 ... 6 ... 30 ... 221
%e 2 ... 5 ... 13 .. 65 .. 145 .. 290 .. 533 .. 2813
%e The following list shows for n = 3 the 2^n purely periodic continued fractions, each followed by the number r it represents, the minimal polynomial a*x^2 + b*x + c of r, the discriminant, D = b^2 - 4*a*c, and the squarefree factor, d, of D.
%e [(1,1,1)] = (1+sqrt(5))/2, -1 - x + x^2, D = 5, d = 5
%e [(1,1,2)] = sqrt(10)/4, -5 + 2 x^2, D = 40, d = 10
%e [(1,2,1)] = (2 + sqrt(10)/3, -2 - 4 x + 3 x^2, D = 10, d = 10
%e [(2,1,1)] = (1 + sqrt(85))/6, -7 - x + 3 x^2, D = 85, d = 85
%e [(1,2,2)] = (1 + sqrt(10)/3, -3 - 2 x + 3 x^2, D = 10, d = 10
%e [(2,1,2)] = (-1 + sqrt(85))/6, -7 + x + 3 x^2, D = 85, d = 85
%e [(2,2,2)] = (5 + sqrt(85))/10, -3 - 5 x + 5 x^2, D = 85, d = 85
%e [(2,2,2)] = sqrt(2), -2 + x^2, D = 8, d = 2
%e The distinct values of d are 2, 5, 10, 85, as in row 3.
%t z = 6;
%t t[n_] := t[n] = Map[FromContinuedFraction[{1, #}] &, Tuples[{1, 2}, n]]
%t u[n_] := u[n] = Table[MinimalPolynomial[t[k], x], {k, 1, n}]
%t d = Discriminant[u[z], x]; v[n_] := Table[{p, m} =
%t Transpose[FactorInteger[k]]; Times @@ (p^Mod[m, 2]), {k, d[[n]]}];
%t w = Table[Union[Table[v[n], {n, 1, z}][[n]]], {n, 1, z}];
%t TableForm[w] (* A246904 array *)
%t Flatten[w] (* A246904 sequence *)
%Y Cf. A246903, A246905.
%K nonn,tabf,easy
%O 1,1
%A _Clark Kimberling_, Sep 06 2014
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