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A206586 Numbers k such that the periodic part of the continued fraction of sqrt(k) has positive even length.

%I A206586 #19 Jan 05 2025 19:51:39
%S A206586 3,6,7,8,11,12,14,15,18,19,20,21,22,23,24,27,28,30,31,32,33,34,35,38,
%T A206586 39,40,42,43,44,45,46,47,48,51,52,54,55,56,57,59,60,62,63,66,67,68,69,
%U A206586 70,71,72,75,76,77,78,79,80,83,84,86,87,88,90,91,92,93,94
%N A206586 Numbers k such that the periodic part of the continued fraction of sqrt(k) has positive even length.
%C A206586 By making the length positive, we exclude squares. See A206587 for this sequence and the squares. All primes of the form 4m + 3 are here.
%H A206586 T. D. Noe, <a href="/A206586/b206586.txt">Table of n, a(n) for n = 1..1000</a>
%H A206586 P. J. Rippon and H. Taylor, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/42-2/quartrippon02_2004.pdf">Even and odd periods in continued fractions of square roots</a>, Fibonacci Quarterly 42, May 2004, pp. 170-180.
%t A206586 Select[Range[100], ! IntegerQ[Sqrt[#]] && EvenQ[Length[ContinuedFraction[Sqrt[#]][[2]]]] &]
%o A206586 (PARI)
%o A206586 cyc(cf) = {
%o A206586   if(#cf==1, return(0)); \\ There is no cycle
%o A206586   my(s=[]);
%o A206586   for(k=2, #cf,
%o A206586     s=concat(s, cf[k]);
%o A206586     if(cf[k]==2*cf[1], return(s)) \\ Cycle found
%o A206586   );
%o A206586   0 \\ Cycle not found
%o A206586 }
%o A206586 select(n->(t=#cyc(contfrac(sqrt(n))))>0 && t%2==0, vector(100, n, n)) \\ _Colin Barker_, Oct 19 2014
%Y A206586 Cf. A003814 (period is odd), A206587.
%K A206586 nonn
%O A206586 1,1
%A A206586 _T. D. Noe_, Mar 19 2012