A257008 Number of Zagier-reduced binary quadratic forms of discriminant n^2 + 4.

1, 2, 3, 5, 5, 10, 7, 13, 14, 16, 12, 31, 13, 24, 29, 38, 17, 44, 26, 47, 46, 34, 30, 90, 34, 56, 49, 63, 39, 106, 40, 87, 77, 70, 57, 139, 55, 58, 89, 149, 52, 138, 52, 136, 123, 92, 69, 223, 84, 104, 146, 111, 62, 218, 94, 214, 121, 132, 96, 296

Offset

1,2

Comments

The number of finite sequences of positive integers with even length parity and alternant equal to n.

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=n^2+4.

The number of possible asymmetry types for the quotient sequence of the odd-length continued fraction expansion of a rational number a/b, where b satisfies one of the congruences b^2 + nb - 1 = 0 (mod a) or b^2 - nb - 1 = 0 (mod a).

Links

Table of n, a(n) for n=1..60.

B. R. Smith, Reducing quadratic forms by kneading sequences, J. Int. Seq., 17 (2014) 14.11.8.

B. R. Smith, End-symmetric continued fractions and quadratic congruences, Acta Arith., 167 (2015) 173-187; also arXiv:1406.7571 [math.NT], 2014.

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

Formula

With D=n^2+4, 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.

Example

For n=4, the a(4) = 5 Zagier-reduced forms of discriminant 20 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, 5*x^2 + 10*x*y + 4*y^2, and 2*x^2 + 6*x*y + 2*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, Map[#^2 + 4 &, Range[3, 60]]}]

Crossrefs

Cf. A257003, A257007, A257009.

Sequence in context: A241381 A237365 A322770 * A338750 A265822 A257012

Adjacent sequences: A257005 A257006 A257007 * A257009 A257010 A257011

Keyword

nonn

Author

Barry R. Smith, Apr 16 2015

Status

approved