A246904 - OEIS

A246904

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.

%I #16 Dec 16 2024 14:33:06

%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

%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 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

%E Edited by _Clark Kimberling_, Dec 05 2024