A033091 -id:A033091 - 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

A033092 Positions of incrementally largest terms in the continued fraction for Euler's constant gamma (A002852).

Original entry on oeis.org

1, 2, 4, 8, 10, 20, 31, 34, 40, 529, 5041, 15347, 25318, 28321, 33261, 158568, 273272, 3233049, 4198630, 11925232, 21988970, 27999430, 130169954, 133517598, 560882701, 1060718271, 1158300012, 1183752952, 3652709607

Offset: 0

Eric W. Weisstein and Bill Gosper

nonn

This sequence assumes nonstandard indexing of c.f. terms as [a_1; a_2, a_3, ...].

No other maximum term occurs in the first 4,851,382,841 terms of the c.f. - Eric W. Weisstein, Jul 22 2013

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

Cf. A224849 (= a(n) - 1).
Cf. A002852 (continued fraction for Euler's constant).
Cf. A033091 (values of incrementally largest terms in c.f.).
Cf. A001620 (decimal expansion of Euler's constant).
Cf. A098967.

a(n) = A224849(n) + 1.

More terms from 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

A098967 Write down decimal expansion of Euler-Mascheroni constant gamma (A001620); divide up into chunks of minimal length so that chunks are increasing numbers and do not begin with 0.

Original entry on oeis.org

5, 7, 72, 156, 649, 1532, 8606, 65120, 90082, 402431, 421593, 3593992, 3598805, 7672348, 8486772, 67776646, 70936947, 632917467, 4951463144, 7249807082, 48096050401, 448654283622, 4173997644923, 5362535003337, 42937337737673

Offset: 0

Sam Handler (shandler(AT)Macalester.edu), Oct 25 2004

			0.57721566490153286060651209008240243104215933593992359880576723488...

G. C. Greubel, Table of n, a(n) for n = 0..1500

Cf. A001620, A002852, A059856, A033091.

f[n_] := Block[{ts = StringDrop[ ToString[ N[n, 250]], 2], a = {}, d = 0, k = 1}, While[ ToExpression[ts] > d, While[d >= ToExpression[ StringTake[ts, k]], k++ ]; te = ToExpression[ StringTake[ts, k]]; d = te; AppendTo[a, te]; ts = StringDrop[ts, k]; If[k > 1, k-- ]]; a]; f[EulerGamma] (* Robert G. Wilson v, Nov 01 2004 *)

Corrected and extended by Robert G. Wilson v, Nov 01 2004

cp's OEIS Frontend

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

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A033092 Positions of incrementally largest terms in the continued fraction for Euler's constant gamma (A002852).

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A098967 Write down decimal expansion of Euler-Mascheroni constant gamma (A001620); divide up into chunks of minimal length so that chunks are increasing numbers and do not begin with 0.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A059856 Write down decimal expansion of Euler-Mascheroni constant gamma (A001620); divide up into minimal chunks so that chunks have increasing length and do not begin with zero.

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Author

Keywords

Examples

Crossrefs

Extensions

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

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Author

Keywords

Comments

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Formula