A267857 Length of the period of the continued fraction for the square root of D, the discriminant of indefinite binary quadratic forms. D is given in A079896.

1, 2, 2, 5, 1, 2, 6, 2, 4, 5, 4, 4, 1, 2, 3, 8, 6, 2, 6, 5, 2, 6, 4, 11, 1, 2, 8, 2, 7, 12, 6, 2, 2, 5, 6, 5, 8, 10, 4, 11, 1, 2, 2, 8, 15, 6, 9, 10, 6, 2, 16, 5, 4, 10, 2, 16, 4, 9, 4, 4, 1, 2, 9, 2, 8, 2, 17, 8, 10, 6, 6, 2, 16, 5, 4, 8, 4, 21

Offset

0,2

Comments

This is a subsequence of A003285.

If a(n) is even then the smallest positive integer solution of the Pell equation x^2 - D(n)*y^2 = +1 with D(n) = A079896(n) is given by (x0, y0) = (P,Q) with P/Q = [a,b[1], ..., b[a(n)-1]. If a(n) is odd then the smallest positive integer solution of the Pell equation x^2 - D(n)*y^2 = +1 is given by (x0, y0) = (P^2 + D(n)*Q^2, 2*P*Q). See e.g., the Silverman reference Theorem 40.4 on p. 351.

For positive integer d, d not a square, the Pell equations X^2 - d*Y^2 = +4 and X^2 - d*Y^2 = -4 have no proper solutions. For D(n) = A079896(n) there are solutions for X^2 - D(n)*Y^2 = +4 or -4 (inclusive or). See the Wolfdieter Lang link under A225953 for Pell +4 or -4 solutions.

References

J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, p. 351.

Links

Table of n, a(n) for n=0..77.

Example

a(0)  = 1  because sqrt(5)  = [2,repeat(4)].
a(1)  = 2  because sqrt(8)  = [2,repeat(1,4)].
a(23) = 11 because sqrt(61) = [7,repeat(1,4,3,1,2 2,1,3,4,1,14)].
Pell +1 equation: n = 23 with D = 61 has odd a(23)
  P/Q = [7,1,4,3,1,2,2,1,3,4,1] = 29718/3805 (in lowest terms). Therefore (x0, y0) = (1766319049, 226153980), see A174762 (Of course, (1, 0) is the smallest nonnegative solution.)

Maple

See the Robert Israel program under A003285, adapted to n -> a(n).

Mathematica

Length[Last@ #] & /@ ContinuedFraction@ Sqrt@ Select[Range@ 200, And[MemberQ[{0, 1}, Mod[#, 4]], ! IntegerQ@ Sqrt@ #] &] (* Michael De Vlieger, Feb 11 2016, after A079896 *)

Crossrefs

Cf. A003285, A033313, A033317, A079896, A225953.

Sequence in context: A011342 A084046 A200493 * A173169 A016586 A073690

Adjacent sequences: A267854 A267855 A267856 * A267858 A267859 A267860

Keyword

nonn,easy

Author

Wolfdieter Lang, Feb 03 2016

Status

approved