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A058281 Continued fraction for square root of e. 10

%I #32 May 05 2022 04:53:28

%S 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 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 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 Continued fraction for square root of e.

%C sqrt(e) = 1.64872127070012814684865078781416357165377610071014801157507... - _Harry J. Smith_, May 01 2009

%H Harry J. Smith, <a href="/A058281/b058281.txt">Table of n, a(n) for n = 0..20000</a>

%H 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 K. Matthews, <a href="http://www.numbertheory.org/php/davison.html">Finding the continued fraction of e^(l/m)</a>

%H T. 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 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_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2,0,0,-1).

%F a(3k+1) = 4k+1, a(i) = 1 otherwise.

%F 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 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

%e sqrt(e) = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(5 + ...)))). - _Harry J. Smith_, May 01 2009

%t ContinuedFraction[ Sqrt[E], 100]

%o (PARI) contfrac(sqrt(exp(1)))

%o (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 Cf. A004766, A016813.

%Y Cf. A019774 for decimal expansion of sqrt(e).

%K cofr,nonn,easy,nice

%O 0,5

%A _Robert G. Wilson v_, Dec 07 2000

%E More terms from _Jason Earls_, Jul 10 2001

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Last modified March 29 04:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)