A257003 - OEIS

A257003

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

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

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OFFSET

0,2

COMMENTS

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.

REFERENCES

D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

LINKS

Barry R. Smith, Table of n, a(n) for n = 0..999

FORMULA

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.

EXAMPLE

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-(n-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.

MATHEMATICA

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) &]}]

CROSSREFS