A006375 Number of equivalence classes of cycles (or periods) of reduced indefinite binary quadratic forms of determinant -n (see comments).

2, 1, 1, 3, 2, 1, 1, 2, 4, 2, 1, 2, 2, 1, 2, 5, 2, 2, 1, 3, 2, 1, 1, 3, 6, 2, 2, 2, 2, 2, 1, 3, 2, 2, 2, 7, 3, 1, 2, 4, 2, 2, 1, 2, 4, 1, 1, 4, 6, 3, 2, 3, 2, 2, 2, 3, 2, 2, 1, 4, 2, 1, 3, 8, 4, 2, 1, 3, 2, 2, 1, 5, 2, 2, 3, 2, 2, 2, 2, 5, 8, 3, 1, 4, 4, 1, 2, 3, 2, 4, 2, 2, 2, 1, 2, 5

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• hard

Comments

From Robin Visser, Jun 05 2025: (Start)

Let a (classically integral) binary quadratic form f(x,y) = a*x^2 + 2*b*x*y + c*y^2 of determinant -n = a*c-b^2 (or equivalently, of discriminant 4*n = 4*(b^2 - a*c)) be denoted as the triple [a,b,c]. If n is not a square, then we can define a sequence of binary quadratic forms [a_0, b_0, c_0], [a_1, b_1, c_1], [a_2, b_2, c_2], ... by the following recursive definition: Let [a_0, b_0, c_0] = [a, b, c], and for each i >= 0, let [a_{i+1}, b_{i+1}, c_{i+1}] = [c_i, t, (t^2 - n)/c_i] where t is the largest integer such that t = -b_i (mod c_i) and t^2 < n, if such an integer t exists. Otherwise t is the smallest integer (in absolute value) which satisfies t = -b_i (mod c_i), taking t positive in the case of a tie (see Conway--Sloane pg 357).

Gauss showed that such sequences are eventually periodic, and we denote the cycle of f(x,y) as the set of all forms in the period of this sequence (see also A087048 for a similar definition of cycle). If n is a square, then this sequence terminates in a form [a_k, b_k, 0], and the definition must be modified slightly (see Conway--Sloane pg 359). Two binary quadratic forms f(x,y) and g(x,y) are said to be properly equivalent if they have the same cycle.

This sequence a(n) counts equivalence classes of such cycles of indefinite binary quadratic forms f(x,y) of determinant -n, with respect to a somewhat coarser notion of equivalence than proper equivalence; here the binary forms [a, b, c], [-a, b, -c], [c, b, a], and [-c, b, -a] are all counted as part of the same equivalence class. (End)

Examples

			From _Robin Visser_, Jun 08 2025: (Start)
For n = 1, every indefinite binary quadratic form of determinant -1 (equivalently discriminant 4) is equivalent to either 2*x*y - y^2 or 2*x*y, thus a(1) = 2.
For n = 2, every indefinite binary quadratic form of determinant -2 (equivalently discriminant 8) is equivalent to x^2 + 2*x*y - y^2, thus a(2) = 1.
For n = 3, every indefinite binary quadratic form of determinant -3 (equivalently discriminant 12) is equivalent to x^2 + 2*x*y - 2*y^2, thus a(3) = 1.
For n = 4, every indefinite binary quadratic form of determinant -4 (equivalently discriminant 16) is equivalent to either x^2 + 2*x*y - 3*y^2, 4*x*y - 2*y^2, or 4*x*y, thus a(4) = 3. (End)
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 362.
• C. F. Gauss, Disquisitiones arithmeticae, Yale University Press, New Haven, Conn.-London, 1966.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.

Crossrefs

Cf. A006371, A006374, A082175, A087048, A256945, A307359.

Programs

SageMath

def a(n):
    S = []
    for b in range(1, ceil(sqrt(n))):
        for a in Integer(n-b^2).divisors():
            c = (b^2-n)/a
            F = [BinaryQF(x,2*b,y) for (x,y) in [(a,c),(-a,-c),(c,a),(-c,-a)]]
            if all([(not Q.is_equivalent(t)) for t in S for Q in F]): S.append(F[0])
    if Integer(n).is_square():
        for c in range(-sqrt(n), sqrt(n)+1):
            F = [BinaryQF(x,2*sqrt(n),y) for (x,y) in [(0,c),(0,-c),(c,0),(-c,0)]]
            if all([(not Q.is_equivalent(t)) for t in S for Q in F]): S.append(F[0])
    return len(S)  # Robin Visser, Jun 06 2025

Extensions

Corrected Apr 15 1995

Name clarified by Robin Visser, May 30 2025

Term a(65) corrected and more terms from Robin Visser, Jun 06 2025