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A131918
Continued fraction expansion of 1 / (1 - gamma - log(3/2)) - 54, where gamma is the Euler-Mascheroni constant.
4
3, 1, 2, 1, 5, 11, 7, 6, 1, 2, 6, 1, 10, 15, 7, 1, 11, 12, 1, 1, 4, 3, 1, 1, 9, 3, 4, 10, 4, 1, 1, 26, 1, 1, 8, 10, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 3, 3, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 4, 2, 1, 49, 7, 1, 2, 1, 1, 2, 16, 1, 283, 1, 1, 5, 1, 1, 1, 2, 1, 30, 19, 1, 11, 2, 5, 10, 3, 1, 4, 1, 6, 2, 19, 1, 1
OFFSET
0,1
COMMENTS
Decimal expansion is A131917.
LINKS
Mark B. Villarino, Ramanujan's Harmonic Number Expansion into Negative Powers of a Triangular Number, arXiv:0707.3950 [math.CA], 2007. Constant occurs in Theorem 7 (DeTemple-Wang), formula (1.14), page 6.
FORMULA
(54 log(3/2) + 54 gamma - 53)/(1 - log(3/2) - gamma) = 1 / (1 - gamma - log(3/2)) - 54, where Martin Fuller simplifies the constant which Villarino showed was implicitly given by DeTemple and Wang.
EXAMPLE
3.73929751945... = 3 + 1/(1 + 1/(2 + 1/(1 + 1/(5 + 1/(11 + ...))))).
MATHEMATICA
ContinuedFraction[1/(1-EulerGamma-Log[3/2])-54, 100] (* Harvey P. Dale, Dec 18 2013 *)
PROG
(PARI) contfrac(1/(1 - Euler - log(3/2)) - 54) \\ Michel Marcus, Mar 11 2013
(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); ContinuedFraction(1/(1 - EulerGamma(R) - Log(3/2)) - 54); // G. C. Greubel, Aug 29 2018
CROSSREFS
Cf. A001008, A001620, A131915, A131916, A131917 (decimal expansion).
Sequence in context: A135261 A339913 A102774 * A010123 A039620 A008296
KEYWORD
cofr,easy,nonn
AUTHOR
Jonathan Vos Post, Jul 27 2007
EXTENSIONS
Offset changed by Andrew Howroyd, Aug 10 2024
STATUS
approved