This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A040002 #58 Aug 10 2025 14:38:11
%S A040002 2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,
%T A040002 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,
%U A040002 4,4,4,4,4,4,4,4,4,4,4,4,4
%N A040002 Continued fraction for sqrt(5).
%C A040002 Decimal expansion of 11/45. - _Natan Arie Consigli_, Jan 19 2016
%D A040002 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 186.
%D A040002 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.
%H A040002 Harry J. Smith, <a href="/A040002/b040002.txt">Table of n, a(n) for n = 0..20000</a>
%H A040002 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>
%H A040002 <a href="/index/Con#confC">Index entries for continued fractions for constants</a>
%H A040002 <a href="/index/Con#constant">Index of eventually constant sequences</a>
%H A040002 <a href="/index/Di#divseq">Index of divisibility sequences</a>
%H A040002 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%F A040002 a(0) = 2, a(n) = 4 n>0. - _Natan Arie Consigli_, Jan 19 2016
%F A040002 From _Elmo R. Oliveira_, Feb 16 2024: (Start)
%F A040002 G.f.: 2*(1+x)/(1-x).
%F A040002 E.g.f.: 4*exp(x) - 2.
%F A040002 a(n) = 2*A040000(n). (End)
%e A040002 2.236067977499789696409173668... = 2 + 1/(4 + 1/(4 + 1/(4 + 1/(4 + ...)))). - _Harry J. Smith_, Jun 01 2009
%p A040002 Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):
%t A040002 ContinuedFraction[Sqrt[5],300] (* _Vladimir Joseph Stephan Orlovsky_, Mar 04 2011 *)
%t A040002 PadRight[{2},120,{4}] (* _Harvey P. Dale_, Jul 06 2019 *)
%o A040002 (PARI) { allocatemem(932245000); default(realprecision, 26000); x=contfrac(sqrt(5)); for (n=0, 20000, write("b040002.txt", n, " ", x[n+1])); } \\ _Harry J. Smith_, Jun 01 2009
%Y A040002 Cf. A002163 (decimal expansion), A001077/A001076 (convergents), A248235 (Egyptian fraction).
%Y A040002 Cf. Continued fraction for sqrt(a^2+1) = (a, 2a, 2a, 2a....): A040000 (contfrac(sqrt(2)) = (1,2,2,...)), A040002, A040006, A040012, A040020, A040030, A040042, A040056, A040072, A040090, A040110 (contfrac(sqrt(122)) = (11,22,22,...)), A040132, A040156, A040182, A040210, A040240, A040272, A040306, A040342, A040380, A040420 (contfrac(sqrt(442)) = (21,42,42,...)), A040462, A040506, A040552, A040600, A040650, A040702, A040756, A040812, A040870, A040930 (contfrac(sqrt(962)) = (31,62,62,...)).
%Y A040002 Essentially the same as A010709.
%K A040002 nonn,cofr,easy,cons
%O A040002 0,1
%A A040002 _N. J. A. Sloane_