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A040006 Continued fraction for sqrt(10).

%I A040006 #66 Jul 26 2025 08:04:15
%S A040006 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 A040006 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 A040006 6,6,6,6,6,6,6,6,6,6,6,6,6
%N A040006 Continued fraction for sqrt(10).
%C A040006 Eventual period is (6). - _Zak Seidov_, Mar 05 2011
%C A040006 The convergents are given in A005667(n+1)/A005668(n+1), n >= 0. - _Wolfdieter Lang_, Nov 23 2017
%C A040006 Decimal expansion of 11/30. - _Elmo R. Oliveira_, Feb 16 2024
%D A040006 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.
%H A040006 Harry J. Smith, <a href="/A040006/b040006.txt">Table of n, a(n) for n = 0..20000</a>
%H A040006 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>
%H A040006 <a href="/index/Con#confC">Index entries for continued fractions for constants</a>
%H A040006 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%F A040006 a(n) = 3 + 3*sign(n). a(n) = 6, n > 0. - _Wesley Ivan Hurt_, Nov 01 2013
%F A040006 From _Elmo R. Oliveira_, Feb 16 2024: (Start)
%F A040006 G.f.: 3*(1+x)/(1-x).
%F A040006 E.g.f.: 6*exp(x) - 3.
%F A040006 a(n) = 3*A040000(n). (End)
%e A040006 3.162277660168379331998893544... = 3 + 1/(6 + 1/(6 + 1/(6 + 1/(6 + ...)))).
%p A040006 Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):
%t A040006 ContinuedFraction[Sqrt[10],300] (* _Vladimir Joseph Stephan Orlovsky_, Mar 04 2011 *)
%t A040006 PadRight[{3},120,{6}] (* _Harvey P. Dale_, Aug 25 2024 *)
%o A040006 (PARI) contfrac(sqrt(10)) \\ For illustration.
%o A040006 (PARI) A040006(n)=if(n,6,3) \\ _M. F. Hasler_, Nov 02 2019
%o A040006 (Magma) [6-3*(Binomial(2*n,n) mod 2): n in [0..100]]; // _Vincenzo Librandi_, Jan 03 2016
%Y A040006 Cf. A010467 (decimal expansion), A005667(n+1)/A005668(n+1) (convergents), A248239 (Egyptian fraction).
%Y A040006 Cf. A040000.
%K A040006 nonn,cofr,easy
%O A040006 0,1
%A A040006 _N. J. A. Sloane_