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A040030 Continued fraction for sqrt(37).

%I A040030 #43 Aug 25 2025 11:11:23
%S A040030 6,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,
%T A040030 12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,
%U A040030 12,12,12,12,12,12,12,12,12
%N A040030 Continued fraction for sqrt(37).
%D A040030 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.
%H A040030 Harry J. Smith, <a href="/A040030/b040030.txt">Table of n, a(n) for n = 0..20000</a>
%H A040030 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>
%H A040030 <a href="/index/Con#confC">Index entries for continued fractions for constants</a>
%H A040030 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%F A040030 From _Elmo R. Oliveira_, Feb 06 2024: (Start)
%F A040030 a(n) = 12 for n >= 1.
%F A040030 G.f.: 6*(1+x)/(1-x).
%F A040030 E.g.f.: 12*exp(x) - 6.
%F A040030 a(n) = 6*A040000(n) = 3*A040002(n) = 2*A040006(n). (End)
%e A040030 6.08276253029821968899968... = 6 + 1/(12 + 1/(12 + 1/(12 + 1/(12 + ...)))). - _Harry J. Smith_, Jun 04 2009
%p A040030 Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):
%t A040030 ContinuedFraction[Sqrt[37],300] (* _Vladimir Joseph Stephan Orlovsky_, Mar 06 2011 *)
%t A040030 PadRight[{6},120,{12}] (* _Harvey P. Dale_, Jan 02 2017 *)
%o A040030 (PARI) { allocatemem(932245000); default(realprecision, 44000); x=contfrac(sqrt(37)); for (n=0, 20000, write("b040030.txt", n, " ", x[n+1])); } \\ _Harry J. Smith_, Jun 04 2009
%Y A040030 Cf. A010491 (decimal expansion), A041060/A041061 (convergents), A248263 (Egyptian fraction).
%Y A040030 Cf. A040000, A040002, A040006, A010851
%K A040030 nonn,cofr,easy,changed
%O A040030 0,1
%A A040030 _N. J. A. Sloane_