A324251 Irregular triangle read by rows: parameters of the principal cycle of discriminant 4*D(n), with D(n) = A000037(n).

-2, 2, -1, 2, -4, 4, -2, 4, -1, 1, -1, 4, -1, 4, -6, 6, -3, 6, -2, 6, -1, 1, -1, 1, -6, 1, -1, 1, -1, 6, -1, 2, -1, 6, -1, 6, -8, 8, -4, 8, -2, 1, -3, 1, -2, 8, -2, 8, -1, 1, -2, 1, -1, 8, -1, 2, -4, 2, -1, 8, -1, 3, -1, 8, -1, 8, -10, 10, -5, 10, -3, 2, -3, 10, -2, 1, -1, 2, -10, 2, -1, 1, -2, 10, -2, 10

Offset: 1

Author

Wolfdieter Lang, Apr 19 2019

Keywords

• sign
• tabf

Comments

The row length of this irregular triangle is 2*A307372(n).

The indefinite binary quadratic Pell form is F = [1, 0, -D(n)], with D(n) = A000037(n) (D not a square). This form is not reduced (see the Buell or Scholz-Schoeneberg references, and the W. Lang link in A225953 for the definition).

The first reduced form, obtained after two equivalence transformations, is FR(n) = [1, 2*s(n), -(D(n) - s(n)^2)] where s(n) = A000194(n) = D(n) - n, for n >= 1. Hence FR(n) = [1, 2*s(n), -(n - s(n)*(s(n)-1))]. For the two transformations invoving R(t) = matrix([0, -1], [1, t]), first with t = 0 then with t = s(n) see a comment in A000194, and the proposition in the W. Lang link given below. FR(n) is the principal form F_p(n) of discriminant 4*D(n).

Each reduced form FR(n) leads to a cycle of reduced forms with (primitive) period P(n) = 2*p(n) = 2*A307372(n). The sequence of R-transformations is given by the parameter tuple (t_1(n), ..., t_{2*p(n)}(n)) with alternating signs which give the row entries T(n, k) = t_k(n). See also Table 2 of the W. Lang link.

The automorphic transformation is obtained by the matrix Auto(n) = R(t_1(n))*R(t_2(n))*...*R(t_{2*p(n)}(n)). Together with the matrix B(n) := R(0)*R(s(n)) = Matrix([-1, s(n)], [0, -1]) one finds all solutions of the Pell equation x^2 - D(n)*y^2 = +1. For each n >= 1 there is one family (also called class) of proper solutions. The general solution is (x(n;j), y(n;j))^T = B(n)*(Auto(n))^j*(1,0)^T, for integer j (T for transposed). One can always choose x >= 1 by an overall sign flip in x and y.

For the general Pell equation x^2 - D(n)*y^2 = N, for integer N, the parallel forms equivalent to FR(n) become important. For details see the W. Lang link given below, section 3.

Examples

			The irregular triangle T(n, k) begins:
n,  D(n) \k   1   2   3   4   5   6   7   8   9  10 ...   2*A324252(n)
----------------------------------------------------------------------
1,   2:      -2   2                                           2
2,   3:      -1   2                                           2
3,   5:      -4   4                                           2
4,   6:      -2   4                                           2
5,   7:      -1   1  -1   4                                   4
6,   8:      -1   4                                           2
7,  10:      -6   6                                           2
8,  11:      -3   6                                           2
9,  12:      -2   6                                           2
10, 13:      -1   1  -1   1  -6   1  -1   1  -1   6          10
11, 14:      -1   2  -1   6                                   4
12, 15:      -1   6                                           2
13, 17:      -8   8                                           2
14, 18:      -4   8                                           2
15, 19:      -2   1  -3   1  -2   8                           6
16, 20:      -2   8                                           2
17, 21:      -1   1  -2   1  -1   8                           6
18, 22:      -1   2  -4   2  -1   8                           6
19, 23:      -1   3  -1   8                                   4
20, 24:      -1   8                                           2
...
--------------------------------------------------------------------
The  forms for the cycle CR(5) for D(5) = 7 (discriminant 28) are:
FR(5) = [1, 4, -3], the transformation with R(-1) produces FR1(5) = [-3, 2, 2], from this R(1) leads to FR2(5) = [2, 2, -3], then with R(-1) to FR3(5) = [-3, 4, 1], and with R(4) back to FR(5).
		

References

• D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, p. 21.
• A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, p. 112.

Links

• Wolfdieter Lang, Cycles of reduced Pell forms, general Pell equations and Pell graphs

Crossrefs

Cf. A000037, A000194, A225953, A307372.

Formula

T(n, k) = t_k(n), the k-th entry of the t-tuple for the R-transformations of the principal cycle for discriminant 4*D(n), with D(n) = A000037(n). See the comments above.