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A013643 Numbers n such that continued fraction for sqrt(n) has period 3. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 29 14:23 EST 2020. Contains 331338 sequences. (Running on oeis4.)