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A058281 Continued fraction for square root of e.

%I A058281 #41 Aug 05 2025 12:57:06
%S A058281 1,1,1,1,5,1,1,9,1,1,13,1,1,17,1,1,21,1,1,25,1,1,29,1,1,33,1,1,37,1,1,
%T A058281 41,1,1,45,1,1,49,1,1,53,1,1,57,1,1,61,1,1,65,1,1,69,1,1,73,1,1,77,1,
%U A058281 1,81,1,1,85,1,1,89,1,1,93,1,1,97,1,1,101,1,1,105,1,1,109,1,1,113,1,1
%N A058281 Continued fraction for square root of e.
%H A058281 Harry J. Smith, <a href="/A058281/b058281.txt">Table of n, a(n) for n = 0..20000</a>
%H A058281 D. H. Lehmer, <a href="/A016825/a016825.pdf">Continued fractions containing arithmetic progressions</a>, Scripta Mathematica, 29 (1973): 17-24. [Annotated copy of offprint]
%H A058281 Keith Matthews, <a href="http://www.numbertheory.org/php/davison.html">Finding the continued fraction of e^(l/m)</a>.
%H A058281 Thomas J. Osler, <a href="http://www.jstor.org/stable/27641838">A proof of the continued fraction expansion of e^(1/M)</a>, Amer. Math. Monthly, 113 (No. 1, 2006), 62-66.
%H A058281 Gang Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>.
%H A058281 <a href="/index/Con#confC">Index entries for continued fractions for constants</a>.
%H A058281 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2,0,0,-1).
%F A058281 a(3k+1) = 4k+1, a(i) = 1 otherwise.
%F A058281 G.f.: -(x^2-x+1)*(x^3-2*x^2-2*x-1) / ((x-1)^2*(x^2+x+1)^2). - _Colin Barker_, Jun 24 2013
%F A058281 E.g.f.: exp(-x/2)*(exp(3*x/2)*(5 + 4*x) + (4 + 8*x)*cos(sqrt(3)*x/2) - 4*sqrt(3)*sin(sqrt(3)*x/2))/9. - _Stefano Spezia_, May 05 2022
%F A058281 Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi + 2*log(sqrt(2)+1)) / (4*sqrt(2)). - _Amiram Eldar_, May 03 2025
%e A058281 sqrt(e) = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(5 + ...)))). - _Harry J. Smith_, May 01 2009
%t A058281 ContinuedFraction[ Sqrt[E], 100]
%t A058281 LinearRecurrence[{0,0,2,0,0,-1},{1,1,1,1,5,1},100] (* _Harvey P. Dale_, Aug 05 2025 *)
%o A058281 (PARI) contfrac(sqrt(exp(1)))
%o A058281 (PARI) { allocatemem(932245000); default(realprecision, 60000); x=contfrac(sqrt(exp(1))); for (n=1, 20001, write("b058281.txt", n-1, " ", x[n])); } \\ _Harry J. Smith_, May 01 2009
%Y A058281 Cf. A004766, A016813.
%Y A058281 Cf. A019774 (decimal expansion of sqrt(e)).
%K A058281 cofr,nonn,easy,nice
%O A058281 0,5
%A A058281 _Robert G. Wilson v_, Dec 07 2000
%E A058281 More terms from _Jason Earls_, Jul 10 2001