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%I M0097 N0034 #86 Sep 08 2022 08:44:31
%S 0,1,1,2,1,2,1,4,3,13,5,1,1,8,1,2,4,1,1,40,1,11,3,7,1,7,1,1,5,1,49,4,
%T 1,65,1,4,7,11,1,399,2,1,3,2,1,2,1,5,3,2,1,10,1,1,1,1,2,1,1,3,1,4,1,1,
%U 2,5,1,3,6,2,1,2,1,1,1,2,1,3,16,8,1,1,2,16,6,1,2,2,1,7,2,1,1,1,3,1,2,1,2
%N Continued fraction for Euler's constant (or Euler-Mascheroni constant) gamma.
%C The first 970258158 terms were computed by Eric Weisstein on Sep 21 2011 using a developmental version of Mathematica.
%C The first 4851382841 terms were computed by Eric Weisstein on Jul 22 2013 using a developmental version of Mathematica.
%C The first 16695279010 terms were computed by _Syed Fahad_ on Apr 29 2021, see link.
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 3.
%D R. S. Lehman, A Study of Regular Continued Fractions. Report 1066, Ballistic Research Laboratories, Aberdeen Proving Ground, Maryland, Feb 1959.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A002852/b002852.txt">Table of n, a(n) for n = 0..10000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H K. Y. Choong, D. E. Daykin and C. R. Rathbone, <a href="https://doi.org/10.1090/S0025-5718-1971-0300981-0 ">Rational approximations to pi</a>, Math. Comp., 25 (1971), 387-392.
%H K. Y. Choong, D. E. Daykin and C. R. Rathbone, <a href="https://doi.org/10.1090/S0025-5718-71-99719-5">Regular continued fractions for pi and gamma</a>, Math. Comp., 25 (1971), 403. [Review of their report at the University of Malaya Computer Centre, September 1970.]
%H Donald E. Knuth, <a href="https://doi.org/10.1090/S0025-5718-1962-0148255-X ">Euler's constant to 1271 places</a>, Math. Comp. 16 (1962), 275-281.
%H Jeffrey C. Lagarias, <a href="http://arxiv.org/abs/1303.1856">Euler's constant: Euler's work and modern developments</a>, arXiv:1303.1856 [math.NT], 2003.
%H Jeffrey C. Lagarias, <a href="https://doi.org/10.1090/S0273-0979-2013-01423-X">Euler's constant: Euler's work and modern developments</a>, Bull. Amer. Math. Soc., 50 (2013), 527-628.
%H Jonathan Sondow, <a href="https://www.dropbox.com/s/2ewoxk79fz512py/AntiSymCE.pdf?dl=0">An antisymmetric formula for Euler's constant</a>, Math. Mag. 71 (1998), 219-220.
%H Jonathan Sondow, <a href="https://doi.org/10.1080/0025570X.1998.11996638">An antisymmetric formula for Euler's constant</a>, Math. Mag. 71 (1998), 219-220.
%H Jonathan Sondow, <a href="http://www.ams.org/journals/proc/2003-131-11/S0002-9939-03-07081-3/S0002-9939-03-07081-3.pdf">Criteria for irrationality of Euler's constant</a>, Proc. Amer. Math. Soc. 131 (2003), 3335-3344.
%H Jonathan Sondow, <a href="http://arXiv.org/abs/math.CA/0211148">Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula</a>, arXiv:math/0211148 [math.CA], 2002-2004.
%H Jonathan Sondow, <a href="https://www.jstor.org/stable/30037385">Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula</a>, Amer. Math. Monthly 112 (2005), 61-65.
%H Jonathan Sondow, <a href="http://arXiv.org/abs/math.CA/0306008">An infinite product for e^gamma via hypergeometric formulas for Euler's constant, gamma</a>, arXiv:math/0306008 [math.CA], 2003.
%H Jonathan Sondow and Sergey Zlobin, <a href="http://arXiv.org/abs/math.NT/0211075">A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant</a>, arXiv:math/0211075 [math.NT], 2002-2009.
%H Jonathan Sondow and Sergey Zlobin, <a href="https://doi.org/10.2478/s12175-009-0127-2">A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant</a>, Math. Slovaca 59 (2009), 1-8.
%H Jonathan Sondow and Wadim Zudilin, <a href="http://arXiv.org/abs/math.NT/0304021">Euler's constant, q-logarithms and formulas of Ramanujan and Gosper</a>, arXiv:math/0304021 [math.NT], 2003.
%H Jonathan Sondow and Wadim Zudilin, <a href="https://doi.org/10.1007/s11139-006-0075-1">Euler's constant, q-logarithms and formulas of Ramanujan and Gosper</a>, Ramanujan J. 12 (2006), 225-244.
%H Syed Fahad, <a href="https://drive.google.com/drive/folders/15LjUnGcZiDAJLptqwl_Wf69zFLV0HkIK?usp=sharing">Euler's Constant - Simple Continued Fraction (16,695,279,010 terms)</a>
%H Syed Fahad, <a href="https://github.com/sinandredemption/zuuv">Zuuv, Multiprecision Floating-point to Continued Fraction Cruncher</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Euler-MascheroniConstant.html">Euler-Mascheroni Constant</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Euler-MascheroniConstantContinuedFraction.html">Euler-Mascheroni Constant Continued Fraction</a>.
%H Gang 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>
%e 0.577215664901532860606512090082402431042...
%e 0 + 1/(1 + 1/(1 + 1/(2 + 1/(1 + 1/(2 + 1/(1 + 1/(4 + 1/(3 + 1/(13 + ...
%t ContinuedFraction[EulerGamma, 100]
%o (PARI) default(realprecision, 11000); x=contfrac(Euler); for (n=0, 10000, write("b002852.txt", n, " ", x[n+1])) \\ _Harry J. Smith_, Apr 14 2009
%o (Magma) ContinuedFraction(EulerGamma(100)); // _Vincenzo Librandi_, Oct 19 2017
%Y Cf. A001620, the decimal expansion, which has many more references.
%Y See also A073004 (exp(gamma)) and A094640 ("alternating Euler constant").
%Y Cf. A033091 (incrementally largest terms), A033092 (positions of incrementally largest terms).
%Y Cf. A033149 (positions of first occurrence of n in the continued fraction).
%K nonn,cofr,nice
%O 0,4
%A _N. J. A. Sloane_
%E More terms from _Robert G. Wilson v_, Dec 08 2000
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