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This is a subsequence of A226166. See the formula.

For details on the cycles for the principal form F_p(n) = FR(n), the first reduced form of the not reduced Pell form F(n) = [1, 0, -D(n)], see A324251, also for references and a W. Lang link with Table 1, last column LCR(n) = 2*a(n).

This is a subsequence of A087048, See the formula.

This sequence is relevant for the Pell forms [1, 0, - D(n)], with D(n) = A000037(n) and discriminant 4*D(n).

The Buell reference, Table 2B, pp. 241-243, gives only the class numbers, called there H, for A000037(n) squarefree and not congruent to 1 modulo 4. E.g., a(3), related to discriminant 4*5 = 20, is not treated there; also a(6) for discriminant 32 = 4*(2*2^2) does not appear there.

For the a(n) cycles of primitive reduced forms of discriminant 4*A000037(n) see the W. lang link in A324251, Table 2 and Table 1, for n = 1..30. - Wolfdieter Lang, Apr 19 2019

This is one of the bisections of sequence A343238. The other sequence is A344232.

This is a proper subsequence of A020669.

The primes in this sequence are given in A033205.

Discriminant Disc = -20 = -4*5. Class number h(-20) = A000003(5) = 2. The reduced primitive forms representing the two proper (determinant = +1) equivalence classes are the present principal form F1 = [1, 0, 5] and F2 = [2, 2, 3] treated in A344232.

A positive integer k is properly represented by some primitive form of Disc = -20 if and only if the congruence s^2 + 20 == 0 (mod 4*k) has a solution. See, e.g., Buell Proposition 41, p. 50, or Scholz-Schoeneberg Satz 74, p. 105. That is, x^2 + 5 == 0 (mod k), with s = 2*x. For the representative solutions x from {0, 1, ..., k-1}, with k from A343238, see A343239. These solutions x determine the so-called representative parallel primitive forms (rpapfs) [k, 2*x, (x^2 + 5)/k] representing k. They are properly equivalent (via so called R(t)-transformations) to one of the reduced forms F1 or F2. (See also W. Lang's links in A225953 and A324251, but there indefinite forms are considered.)

In order to find out which k from A343238 is represented either by form F1 or F2 the two generic multiplicative characters of Disc = -20, namely Legendre(k|p), with the odd prime p = 5 which divides Disc = -20, and Jacobi(-1|k) can be used. See Buell, pp. 51-52. They lead to the two classes of genera of Disc -20.

The present genus I, the principal one, has for odd primes p, not 5, the values Legendre(p|5) = Legendre(5|p) = +1 and Jacobi(-1|p) = Legendre(-1|p) = +1, leading for odd primes not equal to 5 to A033205. The prime 2 is not represented. The prime 5 is trivially represented. For the other genus II these two characters have values -1. There prime 2 is represented.

For composite k the prime number factorization is used, and for powers of primes the lifting theorem is employed (see, e.g., Apostol, p. 121, Theorem 5.30). The solution for prime 2 represented by form F2 = [2, 2, 3] (from the other genus II) is not liftable to powers of 2. The solution for prime 5 is also not liftable (proof by induction). The solutions of the other primes from A033205 and A106865 are uniquely liftable to powers of these primes. See A343238 for all properly represented k for Disc = -20.

For the present genus I the properly represented integers k are given by 2^a*5^b*Product_{j=1..PI} (pI_j)^(eI(j))*Product_{k=1..PII} (pII_k)^(eII(k)), with a and b from {0, 1} but if PI = PII = 0 (empty products are 1) then a = b = 0 giving a(1) = 1. The odd primes pI_j are from A033205 (== {1, 9} (mod 20)), the primes pII_k are from the odd primes of A106865 (== {3, 7}(mod 20)). The exponents of the second product are restricted: if a = 1 then PII >= 1 and Sum_{k=1..PII} eII(k) is odd. If a = 0 then PII >= 0, and if PII >= 1 then this sum is even.

Neighboring numbers k (twins) begin: [5, 6], [29, 30], [45, 46], [69, 70], [205, 206], [229, 230], [245, 246], [269, 270], [405, 406], ...

For the solutions (X, Y) of F2 = [1, 0, 5] properly representing k = a(n) see A344233.

The array A(n, k) gives the number of the representative parallel binary quadratic primitive forms for discriminant Disc(n) = 4*D(n) = 4*A000037(n) and representation of positive integer k which are (properly) equivalent to the Pell form F(n) = [1, 0, -D(n)].

For the definition of representative parallel primitive forms for discriminant Disc > 0 (the indefinte case) and representation of nonzero integers k see the Scholz-Schoeneberg reference, p. 105, or the Buell reference p. 49 (without use of the name parallel). For the procedure to find the primitive representative parallel forms (rpapfs) for Disc(n) = 4*D(n) = 4*A000037(n) and nonzero integer k see the W. Lang link given in A324251, section 3.

Among them the parallel forms which are equivalent to the reduced principal form F_p(n) = [1, 2*s(n), -(D(n) - s(n))^2], with s(n) = A000194(n), are important to find all solutions (x, y) with gcd(x, y) = 1 (proper) of the Pell form F(n) = [1, 0, -D(n)] with Disc(F(n)) = 4*D(n) > 0 representing a positive integer k. The number of these parallel forms pa(n, k) gives the number of the proper fundamental solutions. The general solution is obtained from the fundamental solutions with the help of integer powers of the automorphic matrix corresponding to the cycle determined by the reduced principal form F_p(n).

Thus the array A(n,k) gives the number of proper families (also called classes) of solutions of the Pell equation x^2 - Dn(n)*y^2 = k, for positive integer k. The positions of the nonzero entries in row n give the list of the k values for which proper solutions exist.

These position lists are A057126 (conjecture) and A243655, for k = 1 and 2.

The first column has only 1s, showing that every Pell form [1, 0, -D(n)] represents k = +1, and that there is only one family of proper solutions.

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, ... .

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.

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.

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.

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]].

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.

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.

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.

This is a subset of one half of A082174. See the formula.

This sequence is also one half of the total length of the A307359(n) cycles for discriminant 4*D(n), with D(n) = A000037(n). See the W. Lang link in A324251, Table 2, last column SigmaL(n) = 2*a(n). - Wolfdieter Lang, Apr 19 2019

For details see A324252 which gives the array for the numbers of families of proper solutions of x^2 - D(n)*y^2 = k for positive integers k. See also the W. Lang link in A324251, section 3.

The D(n) values for nonzero entries in column k = 1 are given in A003814 (representation of -1).

The position list for nonzero entries in row n = 1 is A057126 (conjecture).

For the definition of parallel forms for an indefinite binary quadratic form with discriminant Disc and representation of an integer k see, e.g., the Buell, Scholz-Schoeneberg references or the W. Lang link, section 3, with a scanning prescription.

For the Markoff case Disc(n) = 9*m(n) - 4 = b(n)*(b(n)+2), with m = A002559 and b = A324250.

The Markoff form MF(n;x,y) = x^2 - 3*m(n)*x*y + y^2, also written as MF(n) = [1, -3*m(n), 1], representing -m(n)^2, has as first reduced form the principal form F_p(n;X,Y) = X^2 + b(n)*X*Y - b(n)*Y^2, or F_p(n) = [1, b(n), -b(n)], where the connection is X = x-y, Y = x, or x = Y, y = Y - X. Hence X <= 0 for x <= y.

Only proper solutions (gcd(X, Y) = 1) are of interest. Also only primitive representative parallel forms FPa(n;i), for i = 1, 2, ..., #FPa(n), are considered.

In the present case it is possible to give directly the prescription for the primitive representative parallel forms (rpapfs). This is done for the even m(n) == 2 (mod 32) case and the odd case m(n) == 1 (mod 4) separately.

These rpapfs are written as FPa(n;i) = [-m(n)^2, B(n,i), -C(n,i)]. Their number a(n) = #FPa(n) can be found from congruences with an application of the Chinese remainder theorem and the lifting theorem (see Apostol, Theorem 5.26, pp. 118-119, and Theorem 5.30, pp. 121-122 (only part (a) is effective here)). The existence of two solutions for each odd prime modulus is important as input for the lifting to higher prime powers. For each of the singular cases m(1) = 1 and m(2) = 2, without odd prime divisors, there is only one rpapf.

The Frobenius-Markoff uniqueness conjecture is certainly true for m(n) if a(n) = 1 or a(n) = 2. In the latter case the two rpapfs have to be equivalent to the principal form F_p(n), because the known solution implied by the ordered triple MT(n) = (x(n), y(n), m(n)) has an unordered partner solution which after ordering becomes (x(n), y'(n), m(n')) with y'(n) = m(n) and m(n') = 3*x(n)*m(n) - y(n) >= m(n).

See A327344 for details on the congruences which determine the rpapfs.

The corresponding y values are given in A339882.

The Diophantine equation x^2 - p*y^2 = -2, of discriminant Disc = 4*p > 0 (indefinite binary quadratic form), with prime p can have proper solutions (gcd(x, y) = 1) only for primes p = 2 and p == 3 (mod 8) by parity arguments.

There are no improper solutions (with g >= 2, g^2 does not divide 2).

The prime p = 2 has just one infinite family of proper solutions with nonnegative x values. The fundamental proper solutions for p = 2 is (0, 1).

If a prime p congruent to 3 modulo 8, (p(n) = A007520(n)) has a solution then it can have only one infinite family of (proper) solutions with positive x value.

This family is self-conjugate (also called ambiguous, having with each solution (x, y) also (x, -y) as solution). This follows from the fact that there is only one representative parallel primitive form (rpapf), namely F_{pa(n)} = [-2, 2, -(p(n) - 1)/2].

The reduced principal form of Disc(n) = 4*p(n) is F_{p(n)} = [1, 2*s(n), -(p(n) - s(n)^2)], with s(n) = A000194(n'(n)), if p(n) = A000037(n'(n)). The corresponding (reduced) principal cycle has length L(n) = 2*A307372(n'(n)).

The number of all cycles, the class number, for Disc(n) is h(n'(n)) = A324252(n'(n)), Note that in the Buell reference, Table 2B in Appendix 2, p. 241, all Disc(n) <= 4*1051 = 4204 have class number 2, except for p = 443, 499, 659 (Disc = 1772, 1996, 26).

See the W. Lang link Table 1 for some principal reduced forms F_{p(n)} (there for p(n) in the D-column, and F_p is called FR(n)) with their t-tuples, giving the automporphic matrix Auto(n) = R(t_1) R(t_1) ... R(t_{L(n)}), where R(t) := Matrix([[0, -1], [1, t]]), and the length of the principal cycle L(n) given above, and in Table 2 for CR(n).

To prove the existence of a solution one would have to show that the rpapf F{pa(n)} is properly equivalent to the principal form F_{pa(n)}.

This is a subsequence of A237606. There the uninteresting numbers that have improper representations are also recorded.

The primes in the sequence are given in A141302.

A primitive indefinite form F(a,b,c;x,y) = a*x^2 + b*x*y + c*y^2, or [a, b, c] with gcd(a, b, c) = 1 and even discriminant Disc = b^2 - 4*a*c = 60 = 4*D, D = 15, has class number A307359(12) = 4. The four reduced 2-cycle forms are the principal cycle CR = {[1, 6, -6], [6, 6,-1]}, CRhat = {[-1, 6, 6], [-6, 6,1]}, (outer signs flipped), Cy ={[2, 6, -3], [-3, 6, 2]} and Cyhat ={[-2, 6, 3], [3, 6, -2]}.

A proper representation of an integer k (not 0) by such a form F is determined by the rpapfs (representative parallel primitive forms) FPa(k, j) = [k, 2*j, (j^2 - 15)/k], where j from {0, 1, ...,|k|-1} is determined by the congruence j^2 - 15 = = 0 (mod |k|).

The equivalence transformations R(t) of a form F = [a, b, c] is [c , -b +2*c*t, 1 - b*t + c*t^2]. This corresponds to R(t) = Matrix([0, -1], [1, t]). Half-reduced R-transformations use the choice t = ceiling((8 + b)/(2*c) - 1), if c > 0, and t = floor(1 - (8 + b)/(2*|c|)) if c < 0. (c = 0 is not considered because Disc becomes a square).

Because any form F of Disc = 60 represents a negative integer -k if it is equivalent to one of the rpapfs FPa(-k, j), the allowed values are

k = 2^{e_2}*3^{e_3}*5^{e_5}*Product_{i=1..P} p_i^{e_j}, where p_i is an odd prime >= 7 from the sequence A097956 or A038887(n), n >= 4, the p with Legendre(15, p) = +1. The exponents for 2, 3, and 5 are from {0, 1} (these primes are not liftable to powers) and e_i >= 0 (p_i is uniquely liftable to powers, see the Apostol reference), but not all exponents should be 0, because -1 is not represented. The number of infinite families of proper solutions (x, y), with positive values y, is 2^(P).

The present sequence is a proper subset of these generally allowed k values. One has to check if the rpapfs Fpa(-k, j) reach the principal cycle CR, then if so k is a member of the present sequence. This is because the Pell form FPell = [1, 0, -15] reaches (taking t to be first 0 then 3) the cycle member CR(1) = [1, 6, -6], the reduced principal form.

For details see the W. Lang paper in the links.

For the fundamental proper positive solutions of the infinite families for - a(n) see A378711. Note that -a(n) may also have improper solutions besides the proper ones whenever even powers of primes satisfying Legendre(15, p) = +1 appear, e.g., the first instance being -294 = -2*3*7^2).