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 A001204 M4322 N1811 #49 May 04 2025 01:58:26
%S A001204 7,2,1,1,3,18,5,1,1,6,30,8,1,1,9,42,11,1,1,12,54,14,1,1,15,66,17,1,1,
%T A001204 18,78,20,1,1,21,90,23,1,1,24,102,26,1,1,27,114,29,1,1,30,126,32,1,1,
%U A001204 33,138,35,1,1,36,150,38,1,1,39,162,41,1,1,42,174,44,1,1,45,186,47,1,1
%N A001204 Continued fraction for e^2.
%C A001204 Note that e^2 = 7 + 2/(5 + 1/(7 + 1/(9 + 1/(11 + ...)))) (follows from the fact that A004273 is the continued fraction expansion of tanh(1) = (e^2 - 1)/(e^2 + 1)). - _Peter Bala_, Jan 15 2022
%D A001204 Oskar Perron, Die Lehre von den Kettenbrüchen, 2nd ed., Teubner, Leipzig, 1929, p. 138.
%D A001204 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001204 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001204 Harry J. Smith, <a href="/A001204/b001204.txt">Table of n, a(n) for n = 0..20000</a>
%H A001204 Keith Matthews, <a href="http://www.numbertheory.org/php/davison.html">Finding the continued fraction of e^(l/m)</a>.
%H A001204 Gang Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>.
%H A001204 <a href="/index/Con#confC">Index entries for continued fractions for constants</a>.
%H A001204 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,2,0,0,0,0,-1).
%F A001204 G.f.: (x^10 - x^8 - x^7 + x^6 + 4x^5 + 3x^4 + x^3 + x^2 + 2x + 7)/(x^5 - 1)^2. - _Ralf Stephan_, Mar 23 2003
%F A001204 For n > 0, a(5n) = 12n + 6, a(5n+1) = 3n + 2, a(5n+2) = a(5n+3) = 1 and a(5n+4) = 3n + 3. - _Dean Hickerson_, Mar 25 2003
%F A001204 Sum_{n>=5} (-1)^(n+1)/a(n) = (8*sqrt(3)-3)*Pi/72 - 2*log(2)/3. - _Amiram Eldar_, May 04 2025
%e A001204 7.389056098930650227230427460... = 7 + 1/(2 + 1/(1 + 1/(1 + 1/(3 + ...)))).
%t A001204 ContinuedFraction[ E^2, 100]
%t A001204 LinearRecurrence[{0,0,0,0,2,0,0,0,0,-1},{7,2,1,1,3,18,5,1,1,6,30},80] (* _Harvey P. Dale_, Dec 30 2023 *)
%o A001204 (PARI) contfrac(exp(2))
%o A001204 (PARI) allocatemem(932245000); default(realprecision, 95000); x=contfrac(exp(2)); for (n=1, 20001, write("b001204.txt", n-1, " ", x[n])); \\ _Harry J. Smith_, Apr 30 2009
%Y A001204 Cf. A003417, A004273, A005131, A058282, A072334.
%K A001204 nonn,easy,cofr,nice
%O A001204 0,1
%A A001204 _N. J. A. Sloane_
%E A001204 More terms from _Robert G. Wilson v_, Dec 07 2000