A082175 -id:A082175 - cp's OEIS frontend

A082174 Number of primitive reduced indefinite quadratic forms over the integers in two variables with discriminants D(n)=A079896(n).

2, 2, 4, 2, 6, 2, 4, 4, 8, 2, 4, 8, 6, 8, 10, 4, 4, 4, 10, 2, 8, 12, 8, 6, 12, 2, 8, 4, 18, 12, 4, 4, 12, 8, 12, 14, 8, 4, 12, 18, 6, 8, 20, 4, 14, 8, 14, 10, 4, 12, 16, 2, 8, 20, 8, 8, 20, 14, 8, 8, 28, 14, 10, 4, 16, 16, 10, 12, 20, 6, 12, 8, 20, 2, 16, 24, 12, 10, 24, 16, 8, 8, 8, 30

Offset: 1

Wolfdieter Lang, Apr 11 2003

nonn

An indefinite quadratic form in two variables over the integers, a*x^2 + b*x*y + c*y^2 with discriminant D = b^2 - 4*a*c > 0, 0 or 1 (mod 4) and not a square, is called reduced if b>0 and f(D) - min(|2*a|,|2*c|) <= b < f(D), with f(D) := ceiling(sqrt(D)). It is called primitive if gcd(a,b,c)=1 (relative prime). See the Scholz-Schoeneberg reference for these definitions.

			a(1)=2 because there are two reduced forms for D(1)=A079896(1)=5, namely [a,b,c]=[-1, 1, 1] and [1, 1, -1]; here f(5)=3.
a(5)=6: for D(5)=A079896(5)=17 (f(17)=5) the 6 reduced [a,b,c] forms are [[-2, 1, 2], [2, 1, -2], [-2, 3, 1], [-1, 3, 2], [1, 3, -2], [2, 3, -1]]. They are all primitive.
a(6)=2: for D(6)=A079896(6)=20 (f(20)=5) there are four reduced forms: [-2, 2, 2], [2, 2, -2], [-1, 4, 1] and [1, 4, -1], but only two of them are primitive, namely [-1, 4, 1] and [1, 4, -1].

A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, ch.IV, par.31, p. 112 and par.27, p. 97.

Robin Visser, Table of n, a(n) for n = 1..10000
Henri Cohen, A Course in Computational Algebraic Number Theory, Grad. Texts in Math., Vol. 138, Springer-Verlag, Berlin, 1993. xii+534 pp. See Definition 5.6.2 on page 257.

Cf. A082175 (number of reduced forms, nonprimitive forms included).

def a(n):
    i, D, ans = 1, Integer(5), 0
    while(i < n):
        D += 1; i += 1*(((D%4) in [0, 1]) and (not D.is_square()))
    for b in range(1, isqrt(D)+1):
        if ((D-b^2)%4 != 0): continue
        for a in Integer((D-b^2)/4).divisors():
            if (abs(sqrt(D)-2*a)Robin Visser, May 31 2025

a(n)= number of primitive reduced indefinite binary quadratic forms over the integers for D(n)=A079896(n).

Offset corrected and more terms from Robin Visser, May 31 2025

A257003 Number of Zagier-reduced indefinite quadratic forms over the integers in two variables with discriminants D(n) = A079896(n).

Original entry on oeis.org

1, 2, 3, 3, 5, 5, 4, 6, 7, 5, 7, 10, 7, 10, 11, 9, 7, 11, 13, 7, 10, 16, 12, 11, 16, 13, 10, 14, 21, 17, 8, 15, 18, 14, 18, 21, 13, 12, 20, 27, 11, 16, 26, 18, 17, 25, 23, 21, 13, 20, 25, 12, 20, 32, 24, 18, 26, 27, 18, 18, 38, 31, 15, 18, 33

Offset: 1

Barry R. Smith, Apr 14 2015

nonn

An indefinite quadratic form in two variables over the integers, A*x^2 + B*x*y + C*y^2 with discriminant D = B^2 - 4*A*C > 0, 0 or 1 (mod 4) and not a square, is called Zagier-reduced if A>0, C>0, and B>A+C.
This definition is from Zagier's 1981 book, and differs from the older and more common notion of reduced form due to Lagrange (see A082175 for this definition).
The number of pairs of integers (h,k) with |k| < sqrt(D), k^2 congruent to D (mod 4), h > (sqrt(D) - k)/2, h exactly dividing (D-k^2)/4, where D=D(n) is the discriminant being considered.

			For D=20, the pairs (h,k) as above are: (1,4), (2,2), (4,2), (5,0), (4,-2). From these, the a(6)=5 Zagier-reduced forms may be enumerated as h*x^2 + (k+2*h)*x*y + (k+h-(D-k^2)/(4*h))*y^2, yielding x^2+6*x*y+4*y^2, 2*x^2+6*x*y+2*y^2, 4*x^2+10*x*y+5*y^2, 5*x^2+10*x*y+4*y^2, and 4*x^2+6*x*y+y^2.

D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981. See pages 122-123.

Robin Visser, Table of n, a(n) for n = 1..10000 (terms n = 1..1000 from Barry R. Smith).

Cf. A079896, A082175, A257004.

Table[Length[
  Flatten[Select[
    Table[{a, k}, {k,
      Select[Range[Ceiling[-Sqrt[n]], Floor[Sqrt[n]]],
       Mod[# - n, 2] == 0 &]}, {a,
      Select[Divisors[(n - k^2)/4], # > (Sqrt[n] - k)/2 &]}],
    UnsameQ[#, {}] &], 1]], {n,
  Select[Range[
    153], ! IntegerQ[Sqrt[#]] && (Mod[#, 4] == 0 ||
       Mod[#, 4] == 1) &]}]

def a(n):
    i, D, ans = 1, Integer(5), 0
    while(i < n):
        D += 1; i += 1*(((D%4) in [0, 1]) and (not D.is_square()))
    for k in range(-isqrt(D), isqrt(D)+1):
        if ((D-k^2)%4 != 0): continue
        for h in Integer((D-k^2)/4).divisors():
            if h > (sqrt(D) - k)/2: ans += 1
    return ans  # Robin Visser, Jun 01 2025

a(n) equals the number of pairs (h,k) with |k| < sqrt(D), k^2 congruent to D (mod 4), h > (sqrt(D) - k)/2, h exactly dividing (D-k^2)/4, where D=D(n) is the discriminant being considered.

Offset corrected by Robin Visser, Jun 01 2025

A257004 Number of primitive Zagier-reduced indefinite quadratic forms over the integers in two variables with discriminants D(n) = A079896(n).

Original entry on oeis.org

1, 2, 3, 3, 5, 4, 4, 6, 7, 5, 5, 10, 7, 10, 11, 9, 6, 8, 10, 7, 10, 16, 12, 11, 16, 8, 10, 12, 21, 17, 8, 10, 14, 14, 18, 21, 13, 12, 14, 27, 11, 16, 26, 15, 17, 18, 23, 16, 10, 20, 25, 11, 13, 32, 14, 18, 26, 27, 18, 18, 38, 24, 15, 18, 28

Offset: 1

Barry R. Smith, Apr 17 2015

nonn

An indefinite quadratic form in two variables over the integers, A*x^2 + B*x*y + C*y^2 with discriminant D = B^2 - 4*A*C > 0, 0 or 1 (mod 4) and not a square, is called Zagier-reduced if A>0, C>0, and B>A+C.
This definition is from Zagier's 1981 book, and differs from the older and more common notion of reduced form due to Lagrange (see A082175 for this definition).
A form is primitive if its coefficients are relatively prime.

			For D=20, the a(6)=4 Zagier-reduced primitive forms are x^2+6*x*y+4*y^2, 4*x^2+6*x*y+y^2, 4*x^2+10*x*y+5*y^2, and 5*x^2+10*x*y+4*y^2.

D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981. See page 122.

Robin Visser, Table of n, a(n) for n = 1..10000

Cf. A079896, A082174, A257003.

Table[Length[
  Select[Flatten[
    Select[
     Table[{a, k}, {k,
       Select[Range[Ceiling[-Sqrt[n]], Floor[Sqrt[n]]],
        Mod[# - n, 2] == 0 &]}, {a,
       Select[Divisors[(n - k^2)/4], # > (Sqrt[n] - k)/2 &]}],
     UnsameQ[#, {}] &], 1],
   GCD[#[[1]], #[[2]] +
       2*#[[1]], #[[1]] + #[[2]] - (n - #[[2]]^2)/(4*#[[1]])] == 1 &]], {n,
  Select[Range[
    153], ! IntegerQ[Sqrt[#]] && (Mod[#, 4] == 0 || Mod[#, 4] == 1) &]}]

def a(n):
    i, D, ans = 1, Integer(5), 0
    while(i < n):
        D += 1; i += 1*(((D%4) in [0, 1]) and (not D.is_square()))
    for k in range(-isqrt(D), isqrt(D)+1):
        if ((D-k^2)%4 != 0): continue
        for h in Integer((D-k^2)/4).divisors():
            if gcd([h, k+2*h, (k+h-(D-k^2)/(4*h))])==1:
                if h > (sqrt(D)-k)/2: ans += 1
    return ans  # Robin Visser, Jun 01 2025

Offset corrected by Robin Visser, Jun 01 2025

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

Original entry on oeis.org

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

N. J. A. Sloane

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)

			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)

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).

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

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

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

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

cp's OEIS Frontend

A082174 Number of primitive reduced indefinite quadratic forms over the integers in two variables with discriminants D(n)=A079896(n).

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A257003 Number of Zagier-reduced indefinite quadratic forms over the integers in two variables with discriminants D(n) = A079896(n).

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A257004 Number of primitive Zagier-reduced indefinite quadratic forms over the integers in two variables with discriminants D(n) = A079896(n).

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Mathematica

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

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Keywords

Comments

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References

Links

Crossrefs

Programs

SageMath

Extensions