A033149 -id:A033149 - 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

A224847 Position of first occurrence of n in the continued fraction for the Euler-Mascheroni constant (gamma).

Original entry on oeis.org

1, 3, 8, 7, 10, 68, 23, 13, 138, 51, 21, 160, 9, 198, 336, 78, 162, 175, 383, 613, 182, 650, 136, 479, 249, 861, 553, 617, 286, 299, 1951, 165, 149, 2037, 559, 482, 1283, 680, 305, 19, 348, 1129, 2279, 1883, 1902, 2563, 4752, 716, 30, 2609, 567, 247, 2170, 7776

Offset: 1

Eric W. Weisstein, Jul 22 2013

nonn

This sequence is the same as A033149, but uses correct [a_0; a_1, a_2, ...] indexing of continued fraction terms.

The smallest numbers not occurring in the first 4,851,382,841 terms of the c.f. are 27943, 33436, 33978, 34017, ... - Eric W. Weisstein, Jul 22 2013

			The c.f. for gamma is  A002852 = [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, ...].
1 occurs first at term a_1
2 occurs first at term a_3.
3 occurs first at term a_8.
4 occurs first at term a_7.

Eric W. Weisstein, Table of n, a(n) for n = 1..27942
Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant Continued Fraction

Cf. A033149(n) = a(n) + 1.

Cf. A002852 (continued fraction for Euler-Mascheroni constant).

With[{cfeg=Rest[ContinuedFraction[EulerGamma,8000]]},Table[Position[cfeg,n,1,1],{n,60}]]//Flatten (* Harvey P. Dale, Nov 06 2024 *)

cp's OEIS Frontend

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

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