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Continued fraction for sqrt(37).
5

%I #37 Feb 16 2024 12:28:01

%S 6,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,

%T 12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,

%U 12,12,12,12,12,12,12,12,12

%N Continued fraction for sqrt(37).

%H Harry J. Smith, <a href="/A040030/b040030.txt">Table of n, a(n) for n = 0..20000</a>

%H G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>

%H <a href="/index/Con#confC">Index entries for continued fractions for constants</a>

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).

%F From _Elmo R. Oliveira_, Feb 06 2024: (Start)

%F a(n) = 12 for n >= 1.

%F G.f.: 6*(1+x)/(1-x).

%F E.g.f.: 12*exp(x) - 6.

%F a(n) = 6*A040000(n) = 3*A040002(n) = 2*A040006(n). (End)

%e 6.08276253029821968899968... = 6 + 1/(12 + 1/(12 + 1/(12 + 1/(12 + ...)))). - _Harry J. Smith_, Jun 04 2009

%p Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

%t ContinuedFraction[Sqrt[37],300] (* _Vladimir Joseph Stephan Orlovsky_, Mar 06 2011 *)

%t PadRight[{6},120,{12}] (* _Harvey P. Dale_, Jan 02 2017 *)

%o (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 Cf. A010491 (decimal expansion), A041060/A041061 (convergents), A248263 (Egyptian fraction).

%Y Cf. A040000, A040002, A040006.

%K nonn,cofr,easy

%O 0,1

%A _N. J. A. Sloane_