A246904 -id:A246904 - cp's OEIS frontend

A246905 Number of numbers in row n of A246904.

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

Offset: 1

Clark Kimberling, Sep 06 2014

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.

			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.

Cf. A246903, A246904.

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}]]

Edited by Clark Kimberling, Dec 05 2024

A246903 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.

Original entry on oeis.org

5, 8, 5, 8, 12, 5, 8, 40, 85, 5, 8, 12, 96, 221, 480, 5, 8, 145, 260, 533, 1160, 1300, 2813, 5, 8, 12, 40, 85, 672, 1365, 1517, 1680, 3132, 3360, 7565, 16380, 5, 8, 901, 1768, 3725, 3973, 4625, 4901, 7400, 8104, 8468, 8840, 16133, 18229, 39208, 40004, 44104, 95485

Offset: 1

Clark Kimberling, Sep 06 2014

			First 5 rows:
  5 ... 8
  5 ... 8 ... 12
  5 ... 8 ... 40 .. 85
  5 ... 8 ... 12 .. 96 .. 221 . 480
  5 ... 8 ... 145 . 260 . 533 . 1160 . 1300 . 2813
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.
[(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 5, 8, 10, 85, as in row 3.

Cf. A246904, A246905.

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

Edited by Clark Kimberling, Dec 05 2024

A246921 Irregular triangular array: row n gives 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 3s.

Original entry on oeis.org

5, 13, 5, 13, 21, 5, 13, 17, 65, 5, 13, 21, 165, 725, 2805, 5, 13, 445, 1525, 1853, 5933, 7925, 30629, 5, 13, 17, 21, 65, 136, 288, 960, 1260, 4224, 16128, 21608, 83520, 5, 13, 3029, 10205, 11029, 12773, 28157, 34973, 42853, 47965, 53365, 136165, 184045

Offset: 1

Clark Kimberling, Sep 07 2014

			First 5 rows:
  5 ... 13
  5 ... 13 ... 21
  5 ... 13 ... 17 .. 65
  5 ... 13 ... 21 .. 165 .. 725 .. 2805
  5 ... 13 ... 445 . 1525 . 1853 . 5933 . 7925 . 30629
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.
[(1,1,1)] = (1+sqrt(5))/2, -1 - x + x^2, D = 5
[(1,1,3)] = (-1 + sqrt(17))/2, -4 + x + x^2, D = 17
[(1,3,1)] = (3 + sqrt(17))/4, -1 - 3 x + 2 x^2, D = 17
[(3,1,1)] = (1 + sqrt(17))/4, -2 - x + 2 x^2, D = 17
[(1,3,3)] = (-1 + sqrt(65))/4, -8 + x + 2 x^2, D = 65
[(3,1,3)] = (-3 + sqrt(65))/4, -7 + 3 x + 2 x^2, D = 65
[(3,3,1)] = (5 + sqrt(65))/10, -2 - 5 x + 5 x^2, D = 65
[(3,3,3)] = (-1 + sqrt(13))/2, -3 + x + x^2, D = 13
The distinct values of D are 5, 13, 17, 65, as in row 3.

Cf. A246904, A246922.

z = 7; u[n_] := u[n] = Table[MinimalPolynomial[Map[FromContinuedFraction[{1, #}] &, Tuples[{1, 3}, k]], x], {k, 1, n}]; d = Discriminant[u[z], x];
t = Table[Union[d[[n]]], {n, 1, z}]; TableForm[t] (* A246921 array *)
Flatten[t] (* A246921 sequence *)

Edited by Clark Kimberling, Dec 05 2024

A378873 Squarefree part of A378872(n) (the 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, 2, 5, 13, 3, 3, 5, 5, 21, 2, 10, 21, 10, 10, 5, 29, 2, 15, 17, 15, 85, 85, 6, 2, 17, 85, 6, 17, 6, 6, 5, 10, 5, 6, 26, 13, 37, 37, 165, 6, 37, 2, 221, 37, 3, 221, 65, 5, 26, 37, 165, 37, 221, 3, 65, 26, 165, 221, 65, 165, 65, 65, 5, 53, 15, 35, 37, 3, 229

Offset: 1

Pontus von Brömssen, Dec 10 2024

nonn

Any number x whose continued fraction expansion is eventually periodic can be written uniquely as x = (c+f*sqrt(d))/b, where b, c, f, d are integers, b > 0, d > 0 is squarefree, and GCD(b,c,f) = 1. a(n) is equal to d when the periodic part of the continued fraction of x is given by the n-th composition. If two numbers have eventually periodic continued fraction expansions with the same periodic part, their respective values of d are the same.

			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, so a(6) = 3.

Wikipedia, Discriminant of an algebraic number field.

Cf. A007913, A066099 (compositions in standard order), A246904, A246922, A259911, A259912, A305311, A378872, A378874.

a(n) = A007913(A378872(n)) = A378872(n)/A378874(n)^2.
a(2^n) = A259912(n+1) if a(2^n) == 1 (mod 4), a(2^n) = A259912(n+1)/4 otherwise.
For n > k >= 0, a(2^n+2^k) = A259911(n,k+1) if a(2^n+2^k) == 1 (mod 4), a(2^n+2^k) = A259911(n,k+1)/4 otherwise.

cp's OEIS Frontend

A246905 Number of numbers in row n of A246904.

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A246903 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.

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A246921 Irregular triangular array: row n gives 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 3s.

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A378873 Squarefree part of A378872(n) (the discriminant of the minimal polynomial of a number whose continued fraction expansion has periodic part given by the n-th composition (in standard order)).

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