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A010124 Continued fraction for sqrt(19).

%I A010124 #39 Jul 26 2025 09:31:03
%S A010124 4,2,1,3,1,2,8,2,1,3,1,2,8,2,1,3,1,2,8,2,1,3,1,2,8,2,1,3,1,2,8,2,1,3,
%T A010124 1,2,8,2,1,3,1,2,8,2,1,3,1,2,8,2,1,3,1,2,8,2,1,3,1,2,8,2,1,3,1,2,8,2,
%U A010124 1,3,1,2,8,2,1,3,1,2,8,2,1
%N A010124 Continued fraction for sqrt(19).
%D A010124 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.
%H A010124 Harry J. Smith, <a href="/A010124/b010124.txt">Table of n, a(n) for n = 0..20000</a>
%H A010124 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>
%H A010124 A. J. van der Poorten, <a href="https://web.archive.org/web/*/http://www-centre.mpce.mq.edu.au/alfpapers/a075.pdf">An introduction to continued fractions</a>, Unpublished.
%H A010124 A. J. van der Poorten, <a href="/A007400/a007400_4.pdf">An introduction to continued fractions</a>, Unpublished [Cached copy]
%H A010124 <a href="/index/Con#confC">Index entries for continued fractions for constants</a>
%H A010124 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,1).
%F A010124 G.f.: (4 + 2*x + x^2 + 3*x^3 + x^4 + 2*x^5 + 4*x^6)/(1 - x^6). - _Stefano Spezia_, Jul 26 2025
%e A010124 4.358898943540673552236981983... = 4 + 1/(2 + 1/(1 + 1/(3 + 1/(1 + ...)))). - _Harry J. Smith_, Jun 03 2009
%t A010124 ContinuedFraction[Sqrt[19],300] (* _Vladimir Joseph Stephan Orlovsky_, Mar 05 2011 *)
%o A010124 (PARI) { allocatemem(932245000); default(realprecision, 17000); x=contfrac(sqrt(19)); for (n=0, 20000, write("b010124.txt", n, " ", x[n+1])); } \\ _Harry J. Smith_, Jun 03 2009
%Y A010124 Cf. A041028/A041029 (convergents).
%Y A010124 Cf. A010475 (decimal expansion).
%K A010124 nonn,cofr,easy
%O A010124 0,1
%A A010124 _N. J. A. Sloane_