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%I #26 Feb 12 2021 05:47:33
%S 7,14,23,28,32,33,34,47,55,60,62,75,78,79,95,96,98,119,126,128,136,
%T 138,140,141,142,155,167,174,176,180,189,192,194,215,219,220,222,223,
%U 248,252,254,266,287,299,300,305,312,315,318,320,321,322,335,359,368,377,390,392
%N Numbers k such that the continued fraction for sqrt(k) has period 4.
%D Kenneth H. Rosen, Elementary Number Theory and Its Applications, Addison-Wesley, 1984, page 426 (but beware of errors!).
%H T. D. Noe, <a href="/A013644/b013644.txt">Table of n, a(n) for n = 1..1000</a>
%H Mack, Austin and Timothy Sawicki (2012) <a href="http://www.ncurproceedings.org/ojs/index.php/NCUR2012/article/viewFile/195/142">Pell's Equations Through Dynamical Systems</a> [Broken link?]
%F see Mack, Austin and Timothy Sawicki(2012)
%e The continued fraction for sqrt(7) is [2;1,1,1,4,...] with period 4, so 7 is in the sequence. The continued fractions sqrt(3) = [1;1,2,...] with period 2 and sqrt(13) = [3;1,1,1,1,6,...] with period 5 do not have period 4, so 3 and 13 are not in the sequence. - _Michael B. Porter_, Sep 20 2016
%t cfp4Q[n_]:=Module[{sr=Sqrt[n]},!IntegerQ[sr]&&Length[ ContinuedFraction[ sr][[2]]]==4]; Select[Range[500],cfp4Q] (* _Harvey P. Dale_, Jul 29 2014 *)
%Y Cf. A003285.
%K nonn
%O 1,1
%A _Clark Kimberling_
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