This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A246922 #17 Dec 16 2024 14:33:37
%S A246922 5,13,5,13,21,5,13,17,65,5,13,21,29,165,2805,5,13,61,317,445,1853,
%T A246922 5933,30629,2,5,7,13,15,17,21,34,35,65,66,145,5402,5,13,3029,10205,
%U A246922 11029,12773,28157,34973,42853,47965,53365,136165,184045,187493,219965,724205
%N A246922 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 3s.
%e A246922 First 5 rows:
%e A246922 5 ... 13
%e A246922 5 ... 13 ... 21
%e A246922 5 ... 13 ... 17 .. 65
%e A246922 5 ... 13 ... 21 .. 29 ... 165 .. 2805
%e A246922 5 ... 13 ... 61 .. 317 .. 445 .. 1853 .. 5933 .. 30629
%e A246922 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, and the squarefree factor, d, of D.
%e A246922 [(1,1,1)] = (1+sqrt(5))/2, -1 - x + x^2, D = 5
%e A246922 [(1,1,3)] = (-1 + sqrt(17))/2, -4 + x + x^2, D = 17
%e A246922 [(1,3,1)] = (3 + sqrt(17))/4, -1 - 3 x + 2 x^2, D = 17
%e A246922 [(3,1,1)] = (1 + sqrt(17))/4, -2 - x + 2 x^2, D = 17
%e A246922 [(1,3,3)] = (-1 + sqrt(65))/4, -8 + x + 2 x^2, D = 65
%e A246922 [(3,1,3)] = (-3 + sqrt(65))/4, -7 + 3 x + 2 x^2, D = 65
%e A246922 [(3,3,1)] = (5 + sqrt(65))/10, -2 - 5 x + 5 x^2, D = 65
%e A246922 [(3,3,3)] = (-1 + sqrt(13))/2, -3 + x + x^2, D = 13
%e A246922 The distinct values of D are 5, 13, 17, 65, as in row 3. (Here, d = D for all entries, but higher numbered rows, this d < D for some entried.)
%t A246922 z = 8; t[n_] := t[n] = Map[FromContinuedFraction[{1, #}] &, Tuples[{1, 3}, n]]; u[n_] := u[n] = Table[MinimalPolynomial[t[k], x], {k, 1, n}]; d = Discriminant[u[z], x];
%t A246922 v[n_] := Table[{p, m} = Transpose[FactorInteger[k]]; Times @@ (p^Mod[m, 2]), {k, d[[n]]}]; w = Table[Union[Table[v[n], {n, 1, z}][[n]]], {n, 1, z}]; TableForm[w] (* A246922 array *)
%t A246922 Flatten[w] (* A246922 sequence *)
%Y A246922 Cf. A246905, A246921.
%K A246922 nonn,tabf,easy
%O A246922 1,1
%A A246922 _Clark Kimberling_, Sep 07 2014
%E A246922 Edited by _Clark Kimberling_, Dec 05 2024