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

1,1

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

T. D. Noe, Table of n, a(n) for n = 1..200

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

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 *)

Crossrefs

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

Sequence in context: A205797 A203804 A142290 * A142333 A028343 A165816

Adjacent sequences: A013640 A013641 A013642 * A013644 A013645 A013646

Keyword

nonn

Author

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

Status

approved