cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

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.

Original entry on oeis.org

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, 10, 42, 85, 87, 105, 210, 455, 1365, 1517, 7565, 2, 5, 29, 58, 74, 149, 185, 442, 565, 901, 2026, 2117, 2210, 3973, 10001, 11026, 16133, 18229, 2, 3, 5, 6, 26, 30
Offset: 1

Views

Author

Clark Kimberling, Sep 06 2014

Keywords

Comments

Is every squarefree positive integer in this array?

Examples

			First 5 rows:
  2 ... 5
  2 ... 3 ... 5
  2 ... 5 ... 10 .. 85
  2 ... 3 ... 5 ... 6 ... 30 ... 221
  2 ... 5 ... 13 .. 65 .. 145 .. 290 .. 533 .. 2813
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.
[(1,1,1)] = (1+sqrt(5))/2, -1 - x + x^2, D = 5
[(1,1,2)] = sqrt(5/2), -5 + 2 x^2, D = 40
[(1,2,1)] = (2 + sqrt(10))/3, -2 - 4 x + 3 x^2, D = 40
[(2,1,1)] = (1 + sqrt(10))/3, -3 - 2 x + 3 x^2, D = 40
[(1,2,2)] = (1 + sqrt(85))/6, -7 - x + 3 x^2, D = 85
[(2,1,2)] = (-1 + sqrt(85))/6, -7 + x + 3 x^2, D = 85
[(2,2,1)] = (5 + sqrt(85))/10, -3 - 5 x + 5 x^2, D = 85
[(2,2,2)] = sqrt(2), -2 + x^2, D = 8
The distinct values of d are 2, 5, 10, 85, as in row 3.

Crossrefs

Cf. A246903, A246905.

Programs

  • Mathematica
    z = 6;
t[n_] := t[n] = Map[FromContinuedFraction[{1, #}] &, Tuples[{1, 2}, n]]
u[n_] := u[n] = Table[MinimalPolynomial[t[k], x], {k, 1, n}]
d = Discriminant[u[z], x]; 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] (* A246904 array *)
Flatten[w]   (* A246904 sequence *)

Extensions

Edited by Clark Kimberling, Dec 05 2024

A246905 Number of numbers in row n of A246904.

Original entry on oeis.org

2, 3, 4, 6, 8, 13, 18, 30, 46, 78, 125, 220, 374
Offset: 1

Views

Author

Clark Kimberling, Sep 06 2014

Keywords

Comments

Each periodic continued fraction with period an n-tuple of 1s and 2s represents a number r in a quadratic number field, Q(x), where x is a squarefree positive integer, as in A246904, and a(n) = number of distinct numbers x.

Examples

			The following list shows for n = 3 the 2^n 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.
[(1,1,1)] = (1+sqrt(5))/2, -1 - x + x^2, D = 5
[(1,1,2)] = sqrt(5/2), -5 + 2 x^2, D = 40
[(1,2,1)] = (2 + sqrt(10))/3, -2 - 4 x + 3 x^2, D = 40
[(2,1,1)] = (1 + sqrt(10))/3, -3 - 2 x + 3 x^2, D = 40
[(1,2,2)] = (1 + sqrt(85))/6, -7 - x + 3 x^2, D = 85
[(2,1,2)] = (-1 + sqrt(85))/6, -7 + x + 3 x^2, D = 85
[(2,2,1)] = (5 + sqrt(85))/10, -3 - 5 x + 5 x^2, D = 85
[(2,2,2)] = sqrt(2), -2 + x^2, D = 8
The distinct values of d are 2, 5, 10, 85, so that A246905(3) = 4.

Crossrefs

Cf. A246903, A246904.

Programs

  • Mathematica
    z = 7; u[n_] := Table[MinimalPolynomial[Map[FromContinuedFraction[{1, #}] &, Tuples[{1, 2}, k]], x], {k, 1, n}]; Map[Length, Table[Union[Discriminant[u[z], x] [[n]]], {n, 1, z}]]

Extensions

Edited by Clark Kimberling, Dec 05 2024

A378872 Discriminant of the minimal polynomial of a number whose continued fraction expansion has periodic part given by the n-th composition (in standard order).

Original entry on oeis.org

5, 8, 5, 13, 12, 12, 5, 20, 21, 8, 40, 21, 40, 40, 5, 29, 32, 60, 17, 60, 85, 85, 96, 32, 17, 85, 96, 17, 96, 96, 5, 40, 45, 24, 104, 13, 148, 148, 165, 24, 148, 8, 221, 148, 12, 221, 260, 45, 104, 148, 165, 148, 221, 12, 260, 104, 165, 221, 260, 165, 260, 260
Offset: 1

Views

Author

Pontus von Brömssen, Dec 10 2024

Keywords

Comments

Here, the minimal polynomial is required to have integer coefficients with no common divisors.
If two numbers have eventually periodic continued fraction expansions with the same periodic part, the discriminants of their respective minimal polynomials are the same.

Examples

			For n = 6, the 5th composition is (1,2). The value of the continued fraction 1+1/(2+1/(1+1/(2+...))) is (1+sqrt(3))/2, whose minimal polynomial is 2*x^2-2*x-1 with discriminant a(6) = 12.

Crossrefs

Cf. A059893, A066099 (compositions in standard order), A139706, A246903, A246921, A305311, A378873, A378874.

Formula

a(n) = A378873(n)*A378874(n)^2.
a(A059893(n)) = a(n), since reversing the periodic part of a continued fraction leaves the discriminant unchanged.
a(A139706(n)) = a(n), since a circular shift of the periodic part of a continued fraction leaves the discriminant unchanged.
Showing 1-3 of 3 results.

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