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A233592 Positive integers k such that the continued fraction expansion sqrt(k) = c(1) + c(1)/(c(2) + c(2)/(c(3) + c(3)/...)) is periodic.

%I A233592 #22 Jul 12 2021 02:03:29
%S A233592 2,3,5,6,8,10,11,12,15,17,18,20,24,26,27,30,35,37,38,39,40,42,44,45,
%T A233592 48,50,51,56,63,65,66,68,72,80,82,83,84,87,90,99,101,102,104,105,108,
%U A233592 110,120,122,123,132,143,145
%N A233592 Positive integers k such that the continued fraction expansion sqrt(k) = c(1) + c(1)/(c(2) + c(2)/(c(3) + c(3)/...)) is periodic.
%C A233592 For more details on this type of expansion, see A233582.
%C A233592 The cases with aperiodic expansions are listed in A233593.
%C A233592 All the listed cases become periodic after just two leading terms (it is a conjecture that this behavior is general); the validity of their expansions was explicitly tested.
%H A233592 Stanislav Sykora, <a href="/A233592/b233592.txt">Table of n, a(n) for n = 1..200</a>
%H A233592 Stanislav Sykora, <a href="http://dx.doi.org/10.3247/sl4math13.001">Blazys' Expansions and Continued Fractions</a>, Stans Library, Vol.IV, 2013.
%H A233592 Stanislav Sykora, <a href="http://oeis.org/wiki/File:BlazysExpansions.txt">PARI/GP scripts for Blazys expansions and fractions</a>, OEIS Wiki.
%e A233592 Blazys's expansion of sqrt(2) is {1, 2, 4, 4, 4, 4, 4, ...}, i.e., it has a periodic termination. Consequently, 2 is a term of this sequence.
%o A233592 (PARI) See the link.
%Y A233592 Cf. A233593.
%Y A233592 Cf. Blazys's expansions: A233582, A233584, A233585, A233586, A233587.
%K A233592 nonn
%O A233592 1,1
%A A233592 _Stanislav Sykora_, Jan 06 2014