A013643 Numbers k such that the continued fraction for sqrt(k) has period 3.

41, 130, 269, 370, 458, 697, 986, 1313, 1325, 1613, 1714, 2153, 2642, 2834, 3181, 3770, 4409, 4778, 4933, 5098, 5837, 5954, 6626, 7465, 7610, 8354, 9293, 10282, 10865, 11257, 11321, 12410, 13033, 13549, 14698, 14738, 15977, 17266, 17989

Offset: 1

Author

N. J. A. Sloane, Clark Kimberling, and Walter Gilbert

Keywords

• nonn

Comments

All numbers of the form (5n+1)^2 + 4n + 1 for n>0 are elements of this sequence. Numbers of the above form have the continued fraction expansion [5n+1,[2,2,10n+2]]. General square roots of integers with period 3 continued fraction expansions have expansions of the form [n,[2m,2m,2n]]. - David Terr, Jun 15 2004

References

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..200 from T. D. Noe)

Crossrefs

Cf. A044292, A044673, A067896, A028343, A044373, A044754.

Programs

Mathematica

cfp3Q[n_]:=Module[{s=Sqrt[n]},If[IntegerQ[s],1,Length[ ContinuedFraction[ s][[2]]]==3]]; Select[Range[18000],cfp3Q] (* Harvey P. Dale, May 30 2019 *)

Formula

The general form of these numbers is d = d(m, n) = a^2 + 4mn + 1, where m and n are positive integers and a = a(m, n) = (4m^2 + 1)n + m, for which the continued fraction expansion of sqrt(d) is [a;[2m, 2m, 2a]]. - David Terr, Jul 20 2004