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Continued fraction for sqrt(10).
18

%I #62 Sep 27 2024 18:37:25

%S 3,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,

%T 6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,

%U 6,6,6,6,6,6,6,6,6,6,6,6,6

%N Continued fraction for sqrt(10).

%C Eventual period is (6). - _Zak Seidov_, Mar 05 2011

%C The convergents are given in A005667(n+1)/A005668(n+1), n >= 0. - _Wolfdieter Lang_, Nov 23 2017

%C Decimal expansion of 11/30. - _Elmo R. Oliveira_, Feb 16 2024

%H Harry J. Smith, <a href="/A040006/b040006.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 a(n) = 3 + 3*sign(n). a(n) = 6, n > 0. - _Wesley Ivan Hurt_, Nov 01 2013

%F From _Elmo R. Oliveira_, Feb 16 2024: (Start)

%F G.f.: 3*(1+x)/(1-x).

%F E.g.f.: 6*exp(x) - 3.

%F a(n) = 3*A040000(n). (End)

%e 3.162277660168379331998893544... = 3 + 1/(6 + 1/(6 + 1/(6 + 1/(6 + ...)))).

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

%t ContinuedFraction[Sqrt[10],300] (* _Vladimir Joseph Stephan Orlovsky_, Mar 04 2011 *)

%t PadRight[{3},120,{6}] (* _Harvey P. Dale_, Aug 25 2024 *)

%o (PARI) contfrac(sqrt(10)) \\ For illustration.

%o (PARI) A040006(n)=if(n,6,3) \\ _M. F. Hasler_, Nov 02 2019

%o (Magma) [6-3*(Binomial(2*n,n) mod 2): n in [0..100]]; // _Vincenzo Librandi_, Jan 03 2016

%Y Cf. A010467 (decimal expansion), A005667(n+1)/A005668(n+1) (convergents), A248239 (Egyptian fraction).

%Y Cf. A040000.

%K nonn,cofr,easy

%O 0,1

%A _N. J. A. Sloane_