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%I #16 Jun 06 2023 10:50:02
%S -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,
%T 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,
%U -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
%N Irregular triangle read by rows: parameters of the principal cycle of discriminant 4*D(n), with D(n) = A000037(n).
%C The row length of this irregular triangle is 2*A307372(n).
%C 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).
%C 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).
%C 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.
%C 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.
%C 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.
%D D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, p. 21.
%D A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, p. 112.
%H Wolfdieter Lang, <a href="/A324251/a324251_2.pdf">Cycles of reduced Pell forms, general Pell equations and Pell graphs</a>
%F 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.
%e The irregular triangle T(n, k) begins:
%e n, D(n) \k 1 2 3 4 5 6 7 8 9 10 ... 2*A324252(n)
%e ----------------------------------------------------------------------
%e 1, 2: -2 2 2
%e 2, 3: -1 2 2
%e 3, 5: -4 4 2
%e 4, 6: -2 4 2
%e 5, 7: -1 1 -1 4 4
%e 6, 8: -1 4 2
%e 7, 10: -6 6 2
%e 8, 11: -3 6 2
%e 9, 12: -2 6 2
%e 10, 13: -1 1 -1 1 -6 1 -1 1 -1 6 10
%e 11, 14: -1 2 -1 6 4
%e 12, 15: -1 6 2
%e 13, 17: -8 8 2
%e 14, 18: -4 8 2
%e 15, 19: -2 1 -3 1 -2 8 6
%e 16, 20: -2 8 2
%e 17, 21: -1 1 -2 1 -1 8 6
%e 18, 22: -1 2 -4 2 -1 8 6
%e 19, 23: -1 3 -1 8 4
%e 20, 24: -1 8 2
%e ...
%e --------------------------------------------------------------------
%e The forms for the cycle CR(5) for D(5) = 7 (discriminant 28) are:
%e 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).
%Y Cf. A000037, A000194, A225953, A307372.
%K sign,tabf
%O 1,1
%A _Wolfdieter Lang_, Apr 19 2019
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