The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A257003 Number of Zagier-reduced 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 (list; graph; refs; listen; history; text; internal format)
 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 31 13:01 EDT 2020. Contains 334748 sequences. (Running on oeis4.)