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: 1

Author

Wolfdieter Lang, Feb 03 2016

Keywords

• nonn
• easy

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.

Examples

			a(1)  = 1  because sqrt(5)  = [2,repeat(4)].
a(2)  = 2  because sqrt(8)  = [2,repeat(1,4)].
a(24) = 11 because sqrt(61) = [7,repeat(1,4,3,1,2,2,1,3,4,1,14)].
Pell +1 equation: n = 24 with D = 61 has odd a(24)
  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.)
		

References

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

Links

• Robin Visser, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

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 *)

SageMath

def a(n):
    i, D = 1, Integer(5)
    while(i < n):
        D += 1; i += 1*(((D%4) in [0, 1]) and (not D.is_square()))
    K. = QuadraticField(D)
    return continued_fraction(a).period_length()  # Robin Visser, Jun 06 2025

Extensions

Offset corrected by Robin Visser, Jun 06 2025