A131918 Continued fraction expansion of 1 / (1 - gamma - log(3/2)) - 54, where gamma is the Euler-Mascheroni constant.

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

1,1

Comments

Decimal expansion is A131917. Abstract: An algebraic transformation of the DeTemple-Wang half-integer approximation to the harmonic series produces the general formula and error estimate for the Ramanujan expansion for the n-th harmonic number into negative powers of the n-th triangular number. We also discuss the history of the Ramanujan expansion for the n-th harmonic number as well as sharp estimates of its accuracy, with complete proofs, and we compare it with other approximative formulas.

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

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+ 1/7+ 1/6+ 1/1+ 1/2+ 1/6+ 1/1+ 1/10+ 1/15+ 1/7+ 1/1+ 1/11+ 1/12+ 1/1+ 1/1+ 1/4+ 1/3+ 1/1+ 1/1+ 1/9+ 1/3+ 1/4+ 1/10+ 1/4+ 1/1+ 1/1+ 1/26+ 1/1+ 1/1+ 1/8+ 1/10+ 1/1+ 1/2+ 1/1+ 1/1+ 1/1+ 1/2+ 1/2+ 1/1+ 1/1+ 1/3+ 1/1+ 1/3+ 1/3+ 1/1+ 1/1+ 1/1+ 1/2+ 1/1+ 1/1+ 1/3+ 1/1+ 1/1+ 1/1+ 1/2+ 1/4+ 1/2+ 1/1+ 1/49+ 1/7+ 1/1+ 1/2+ 1/1+ 1/1+ 1/2+ 1/16+ 1/1+ 1/283+ 1/1+ 1/1+ 1/5+ 1/1+ 1/1+ 1/1+ 1/2+ 1/1+ 1/30+ 1/19+ 1/1+ 1/11+ 1/2+ 1/5+ 1/10+ 1/3+ 1/1+ 1/4+ 1/1+ 1/6+ 1/2+ 1/19+ 1/1+ 1/1+ 1/3+ 1/2+ 1/1+ ...

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.

Sequence in context: A135261 A339913 A102774 * A010123 A039620 A008296

Adjacent sequences: A131915 A131916 A131917 * A131919 A131920 A131921

Keyword

cofr,easy,nonn

Author

Jonathan Vos Post, Jul 27 2007

Status

approved