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%I #14 Feb 03 2023 03:17:12
%S 1,2,9,18,21,29,37,42,53,57,58,74,81,93,106,109,113,114,133,137,141,
%T 149,162,177,186,189,193,197,217,218,226,233,249,261,266,274,277,281,
%U 282,298,309,317,329,333,337,354,361,373,378,386,389,393,394,401,413,417,421,434,449,457,466,477,498,501
%N Positive integers that are properly represented by each primitive binary quadratic form of discriminant 28 that is properly equivalent to the principal form [1, 4, -3].
%C This is a subsequence of A242662, excluding the primitive forms of discriminant 28 with only improper representations of k, like k = 4, 8, 16, 25, 32, ... .
%C An indefinite binary quadratic primitive form F = a*x^2 + b*x*y + c*y^2 (gcd(a, b, c) = 1) with discriminant Disc = b^2 - 4*a*c = 28 = 2^2*7 is denoted by [a, b, c], or in matrix notation by MF = Matrix([[a, b/2], [b/2, c]]). Hence F = X*MF*X^T (T for transposed), where X = (x, y). See the two links for details and references.
%C Properly equivalent forms F' and F are related by a matrix R of determinant +1 like MF' = R^T*MF*R, and X'^T = R^{-1}*X^T.
%C Each primitive form, properly equivalent to the reduced principal form F_p = [1, 4, -3] for Disc = 28 (used in A242662), represents the given nonnegative k = a(n) values (and only these) properly with X = (x, y) and gcd(x, y) = 1. Modulo an overall sign change in X one can choose x nonnegative.
%C There are 8 = A082174(8) primitive reduced forms of Disc = 28 leading to 2 = A087048(8) (class number) cycles each of period 4, namely the principal cycle CyP = [[1, 4, -3], [-3, 2, 2], [2, 2, -3], [-3, 4, 1]] and the one (with outer signs flipped) CyP' = [[-1, 4, 3], [3, 2, -2], [-2, 2, 3], [3, 4, -1]].
%C There are A358947(n) representative parallel primitive forms (rpapfs) of discriminant Disc = 28 for k = a(n). This gives the number of proper fundamental representations X = (x, y), with x >= 0, of each primitive form [a, b, c], properly equivalent to the principal form F_p of Disc = 28.
%C For the negative integers k properly represented by primitive forms [a, b, c] properly equivalent to the principal form of Disc = 28 see A359476. The corresponding number of fundamental proper representations is given in A359477.
%C This and the three related sequences originated from a proposal by _Klaus Purath_ proving that the form FKP := [1, -2, -6] of Disc = 28 represents k = k(m) = m^2 - 7 = A028881(m), for m >= 3, with the two fundamental representations X1(m) = (m+1, 1) and X2(m) = (11*m - 29, 3*m - 8). This form FKP is properly equivalent to the principal form F_p with R = Matrix([[1, -3], [0, 1]]). Hence all k = a(n) are represented by the form FKP, and A028881 is a subsequence of the present one.
%H Wolfdieter Lang, <a href="/A225953/a225953_3.pdf">Periods of Indefinite Binary Quadratic Forms, Continued Fractions and the Pell +/-4 Equations.</a>
%H Wolfdieter Lang, <a href="/A324251/a324251_2.pdf">Cycles of reduced Pell forms, general Pell equations and Pell graphs</a>
%e k = 9 = a(3): F = FPell = [1, 0, -7] is properly equivalent to F_p = [1, 4, -3] by two so-called half-reduced right neighbor R(t)-transformations, with the matrix R = R(t) = Matrix([[0, -1], [1, t]]), first with t = 0 then with t = 2. For FPell representing k = 9 with x > 0 and y > 0 see X_1(9, i) = (A307168(i), A307169(i)) and X_2(9, i) = (A307172(i), A307173(i)), for i >= 0. There are also the representations with y -> -y arising from the opposite fundamental solutions.
%e The 2 = A358947(3) rpapfs are F1 = [9, 8, 1] and F2 = [9, 10, 2]. They lead by proper equivalence transformations to a form of the above given principal cycle CyP. F1 -> [1, 4, -3] = F_p with matrix R(6), and F2 -> [2, 2, -3] with R(3). See the FIGURE, p. 10, of the linked paper.
%e Besides the primitive forms FPell, F1, F2 and the four forms of CyP also F' = [-7, 0, 1], and all primitive and properly equivalent forms represent k = 9. See the mentioned FIGURE, where FPa1 = F1, FPa1 = F2, Fpa2' = F_p^{(2)} = [2, 2, -3] and FPa2'' = F_p^{(3)} = [-3, 4, 1].
%Y Cf. A028881, A242662, A307168, A307169, A307172, A307173, A358947, A359476, A359477.
%K nonn
%O 1,2
%A _Wolfdieter Lang_, Jan 10 2023
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