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
Keywords
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)
Programs
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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
Comments