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

1,1

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

Vincenzo Librandi, Table of n, a(n) for n = 1..2001

S. R. Finch, Class number theory

Steven R. Finch, Class number theory [Cached copy, with permission of the author]

Formula

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

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

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A076635 A294227 A116602 * A133315 A003658 A003656

Adjacent sequences: A079893 A079894 A079895 * A079897 A079898 A079899

Keyword

nonn,easy

Author

Wolfdieter Lang, Jan 31 2003

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

Status

approved