A079896 Discriminants of indefinite binary quadratic forms.

5, 8, 12, 13, 17, 20, 21, 24, 28, 29, 32, 33, 37, 40, 41, 44, 45, 48, 52, 53, 56, 57, 60, 61, 65, 68, 69, 72, 73, 76, 77, 80, 84, 85, 88, 89, 92, 93, 96, 97, 101, 104, 105, 108, 109, 112, 113, 116, 117, 120, 124, 125, 128, 129, 132, 133, 136, 137, 140, 141, 145, 148

Offset: 1

Author

Wolfdieter Lang, Jan 31 2003

Keywords

• nonn
• easy

Comments

Numbers n such that n == 0 (mod 4) or n == 1 (mod 4), but n is not a square.

For an indefinite binary quadratic form over the integers a*x^2 + b*x*y + c*y^2 the discriminant is D = b^2 - 4*a*c > 0; and D not a square is assumed.

Also, a superset of A227453. - Ralf Stephan, Sep 22 2013

For the period length of the continued fraction of sqrt(a(n)) see A267857(n). - Wolfdieter Lang, Feb 18 2016

[I changed the offset to 1, since this is an important list. Many parts of the entry, including the b-file, will need to be changed. - N. J. A. Sloane, Mar 14 2023]

References

• McMullen, Curtis. "Billiards and Teichmüller curves." Bulletin of the American Mathematical Society, 60:2 (2023), 195-250. See Table C.1.
• A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, p. 112.

Links

• Robin Visser, Table of n, a(n) for n = 1..10000 (terms n = 1..2001 from Vincenzo Librandi).
• S. R. Finch, Class number theory
• Steven R. Finch, Class number theory [Cached copy, with permission of the author]

Crossrefs

Cf. A014601, A042948 (with squares), A087048 (class numbers), A267857.

Programs

Mathematica

Select[ Range[148], (Mod[ #, 4] == 0 || Mod[ #, 4] == 1) && !IntegerQ[ Sqrt[ # ]] & ]

PARI

seq(N) = {
  my(n = 1, v = vector(N), top = 0);
  while (top < N,
    if (n%4 < 2 && !issquare(n), v[top++] = n); n++;);
  return(v);
};
seq(62) \\ Gheorghe Coserea, Nov 07 2016

Formula

a(2*k^2 + 2*k + 1) = 4*(k+1)^2 + 1 for k >= 0. - Gheorghe Coserea, Nov 07 2016

a(2*k^2 + 4*k + 2 + (k+1)*(-1)^k) = (2*k + 3)*(2*k + 3 + (-1)^k) for k >= 0. - Bruno Berselli, Nov 10 2016

Extensions

More terms from Robert G. Wilson v, Mar 26 2003

Offset changed to 1 (since this is a list). - N. J. A. Sloane, Mar 14 2023