A013642 Numbers k such that the continued fraction for sqrt(k) has period 2.

3, 6, 8, 11, 12, 15, 18, 20, 24, 27, 30, 35, 38, 39, 40, 42, 48, 51, 56, 63, 66, 68, 72, 80, 83, 84, 87, 90, 99, 102, 104, 105, 110, 120, 123, 132, 143, 146, 147, 148, 150, 152, 156, 168, 171, 182, 195, 198, 200, 203, 210, 224, 227, 228, 230, 231, 235, 240, 255, 258, 260, 264

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• changed

Comments

This sequence is identical to the sequence of numbers of the form k = a^2 + b, where a and b are positive integers and b is a factor of 2a greater than 1, in which case the continued fraction expansion of sqrt(k) is [a; [2a/b, 2a]]. - David Terr, Jun 11 2004

References

• Kenneth H. Rosen, Elementary Number Theory and Its Applications, Addison-Wesley, 1984, page 426 (but beware of errors!).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Crossrefs

Cf. A001066, A043549, A026604, A047219, A090848, A004957.

Programs

Mathematica

cf2Q[n_]:=Module[{s=Sqrt[n]},If[IntegerQ[s],1,Length[ ContinuedFraction[ s][[2]]]]==2]; Select[Range[300],cf2Q] (* Harvey P. Dale, Jun 21 2017 *)