A033092 -id:A033092 - cp's OEIS frontend

A002852 Continued fraction for Euler's constant (or Euler-Mascheroni constant) gamma.

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, 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, 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

Offset: 0

N. J. A. Sloane

The first 970258158 terms were computed by Eric W. Weisstein on Sep 21 2011 using a developmental version of Mathematica.
The first 4851382841 terms were computed by Eric W. Weisstein on Jul 22 2013 using a developmental version of Mathematica.
The first 16695279010 terms were computed by Syed Fahad on Apr 29 2021, see link.

			0.577215664901532860606512090082402431042...
0 + 1/(1 + 1/(1 + 1/(2 + 1/(1 + 1/(2 + 1/(1 + 1/(4 + 1/(3 + 1/(13 + ...

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.
R. S. Lehman, A Study of Regular Continued Fractions. Report 1066, Ballistic Research Laboratories, Aberdeen Proving Ground, Maryland, Feb 1959.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
K. Y. Choong, D. E. Daykin and C. R. Rathbone, Rational approximations to pi, Math. Comp., 25 (1971), 387-392.
K. Y. Choong, D. E. Daykin and C. R. Rathbone, Regular continued fractions for pi and gamma, Math. Comp., 25 (1971), 403. [Review of their report at the University of Malaya Computer Centre, September 1970.]
Donald E. Knuth, Euler's constant to 1271 places, Math. Comp. 16 (1962), 275-281.
Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, arXiv:1303.1856 [math.NT], 2003.
Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, Bull. Amer. Math. Soc., 50 (2013), 527-628.
Jonathan Sondow, An antisymmetric formula for Euler's constant, Math. Mag. 71 (1998), 219-220.
Jonathan Sondow, An antisymmetric formula for Euler's constant, Math. Mag. 71 (1998), 219-220.
Jonathan Sondow, Criteria for irrationality of Euler's constant, Proc. Amer. Math. Soc. 131 (2003), 3335-3344.
Jonathan Sondow, Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula, arXiv:math/0211148 [math.CA], 2002-2004.
Jonathan Sondow, Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula, Amer. Math. Monthly 112 (2005), 61-65.
Jonathan Sondow, An infinite product for e^gamma via hypergeometric formulas for Euler's constant, gamma, arXiv:math/0306008 [math.CA], 2003.
Jonathan Sondow and Sergey Zlobin, A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant, arXiv:math/0211075 [math.NT], 2002-2009.
Jonathan Sondow and Sergey Zlobin, A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant, Math. Slovaca 59 (2009), 1-8.
Jonathan Sondow and Wadim Zudilin, Euler's constant, q-logarithms and formulas of Ramanujan and Gosper, arXiv:math/0304021 [math.NT], 2003.
Jonathan Sondow and Wadim Zudilin, Euler's constant, q-logarithms and formulas of Ramanujan and Gosper, Ramanujan J. 12 (2006), 225-244.
Syed Fahad, Euler's Constant - Simple Continued Fraction (16,695,279,010 terms)
Syed Fahad, Zuuv, Multiprecision Floating-point to Continued Fraction Cruncher
Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant.
Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant Continued Fraction.
Gang Xiao, Contfrac.
Index entries for continued fractions for constants

Cf. A001620, the decimal expansion, which has many more references.
See also A073004 (exp(gamma)) and A094640 ("alternating Euler constant").
Cf. A033091 (incrementally largest terms), A033092 (positions of incrementally largest terms).
Cf. A033149 (positions of first occurrence of n in the continued fraction).

ContinuedFraction(EulerGamma(100)); // Vincenzo Librandi, Oct 19 2017

Mathematica
```
ContinuedFraction[EulerGamma, 100]
```

default(realprecision, 11000); x=contfrac(Euler); for (n=0, 10000, write("b002852.txt", n, " ", x[n+1])) \\ Harry J. Smith, Apr 14 2009

More terms from Robert G. Wilson v, Dec 08 2000

A033091 Incrementally largest terms in the continued fraction for Euler's constant gamma (A002852).

Original entry on oeis.org

0, 1, 2, 4, 13, 40, 49, 65, 399, 2076, 11626, 12156, 12190, 16992, 87983, 717895, 2186175, 4327480, 7928565, 8130547, 12139891, 54517279, 67728780, 264656137, 410041022, 2970688427, 13126642049, 19585729892, 71713995073

Offset: 0

Eric W. Weisstein and Bill Gosper

a(28)=71713995073 is the largest of the first 4851382841 terms of the c.f. - Eric W. Weisstein, Jul 22 2013

a(28)=71713995073 is the largest of the first 16695279010 terms of the c.f. - Syed Fahad, May 05 2021

Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant Continued Fraction

Cf. A002852 (continued fraction for Euler's constant).
Cf. A033092 (positions of incrementally largest terms in c.f.).
Cf. A001620 (decimal expansion of Euler's constant).
Cf. A098967.
Cf. A059856, A098967.

More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com) and (independently) Eric W. Weisstein, Oct 25 2004
More terms from Eric W. Weisstein, Jan 02 2007
a(22) and a(23) from Eric W. Weisstein, Dec 09 2010
a(24) from Eric W. Weisstein, Sep 21 2011
a(25)-a(28) from Eric W. Weisstein, Jul 22 2013

A033090 Indices of incrementally largest terms in the continued fraction for Pi.

Original entry on oeis.org

1, 2, 3, 5, 308, 432, 28422, 156382, 267314, 453294, 11504931, 849955263, 2349980289, 3588031780, 8600404591, 15621034283

Offset: 1

Eric W. Weisstein, Bill Gosper

This sequence assumes nonstandard indexing of continued fraction terms as [a_1; a_2, a_3, ...]. If you use the actual offset from A001203, corresponding to [a_0; a_1, a_2, ...], you get instead 0, 1, 2, 4, 307, 431, 28421, ... Compare with A033092 versus A224849. - Jeppe Stig Nielsen, Dec 14 2019

Syed Fahad, 30 billion terms of the simple continued fraction of Pi
E. Fontich, C. Simó, and A. Vieiro, On the "hidden" harmonics associated to best approximants due to quasiperiodicity in splitting phenomena, Regular and Chaotic Dynamics (2018), Pleiades Publishing, Vol. 23, Issue 6, 638-653. Also PDF.
Eric Weisstein's World of Mathematics, Pi
Eric Weisstein's World of Mathematics, Pi Continued Fraction

Cf. A033089, A001203.

With[{s = ContinuedFraction[Pi, 2*10^7]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Jan 31 2020 *)

a(12) from Eric W. Weisstein, Dec 08 2010
a(13) from Eric W. Weisstein, Sep 16 2011
a(14) from Eric W. Weisstein, Sep 17 2011
a(15) from Eric W. Weisstein, Jul 18 2013
a(16) from Syed Fahad, Apr 27 2021

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A002852 Continued fraction for Euler's constant (or Euler-Mascheroni constant) gamma.

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A033091 Incrementally largest terms in the continued fraction for Euler's constant gamma (A002852).

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A033090 Indices of incrementally largest terms in the continued fraction for Pi.

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A224849 Positions of incrementally largest terms in the continued fraction for Euler's constant gamma (A002852).

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