

A257003


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


7



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 Zagierreduced 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 (Dk^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 (Dk^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 Zagierreduced forms may be enumerated as h*x^2 + (k+2*h)*x*y + (k+h(nk^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

Cf. A079896, A082175, A257004.
Sequence in context: A273493 A193404 A072923 * A131922 A260718 A113730
Adjacent sequences: A257000 A257001 A257002 * A257004 A257005 A257006


KEYWORD

nonn


AUTHOR

Barry R. Smith, Apr 14 2015


STATUS

approved



