

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
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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, AddisonWesley, 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



