A131916 Continued fraction expansion of 1 / (1 - gamma - log(sqrt(2))) - 12, where gamma is the Euler-Mascheroni constant.

1, 8, 4, 2, 1, 5, 1, 1, 5, 1, 1, 3, 8, 1, 4, 2, 6, 9, 5, 97, 2, 1, 1, 2, 9, 8, 5, 3, 16, 1, 31, 2, 1, 8, 7, 1, 2, 1, 15, 1, 1, 1, 3, 1, 12, 3, 11, 2, 1, 1, 13, 1, 3, 6, 1, 15, 9, 3, 4, 1, 2, 1, 5, 1, 1, 1, 18, 2, 5, 8, 1, 3, 1, 1, 2, 3, 12, 1, 8, 1, 2, 1, 1, 1, 2, 17, 13, 3, 1, 1, 1, 5, 1, 1, 3, 2, 1, 2, 3

Offset

1,2

Comments

Decimal expansion is A131915.

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 6, formula (1.13), page 6.

Formula

Martin Fuller corrected a typo in the cited paper. It should be: ((12 * gamma) - 11 + (12 * log(2^0.5))) / (1 - gamma - log(2^0.5)) or more simply: 1 / (1 - gamma - log(2^0.5)) - 12.

Example

1.1215093407... = 1 + 1/8+ 1/4+ 1/2+ 1/1+ 1/5+ 1/1+ 1/1+ 1/5+ 1/1+ 1/1+ 1/3+ 1/8+ 1/1+ 1/4+ 1/2+ 1/6+ 1/9+ 1/5+ 1/97+ 1/2+ 1/1+ 1/1+ 1/2+ 1/9+ 1/8+ 1/5+ 1/3+ 1/16+ 1/1+ 1/31+ 1/2+ 1/1+ 1/8+ 1/7+ 1/1+ 1/2+ 1/1+ 1/15+ 1/1+ 1/1+ 1/1+ 1/3+ 1/1+ 1/12+ 1/3+ 1/11+ 1/2+ 1/1+ 1/1+ 1/13+ 1/1+ 1/3+ 1/6+ 1/1+ 1/15+ 1/9+ 1/3+ 1/4+ 1/1+ 1/2+ 1/1+ 1/5+ 1/1+ 1/1+ 1/1+ 1/18+ 1/2+ 1/5+ 1/8+ 1/1+ 1/3+ 1/1+ 1/1+ 1/2+ 1/3+ 1/12+ 1/1+ 1/8+ 1/1+ 1/2+ 1/1+ 1/1+ 1/1+ 1/2+ 1/17+ 1/13+ 1/3+ 1/1+ 1/1+ 1/1+ 1/5+ 1/1+ 1/1+ 1/3+ 1/2+ 1/1+ 1/2+ 1/3+ 1/5+ . . .

Mathematica

ContinuedFraction[1/(1-EulerGamma-Log[Sqrt[2]])-12, 100] (* Harvey P. Dale, Oct 01 2015 *)

Prog

(PARI) contfrac(1/(1 - Euler - log(sqrt(2))) - 12) \\ Michel Marcus, Mar 11 2013
(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); ContinuedFraction(1/(1 - EulerGamma(R) - Log(Sqrt(2))) - 12); // G. C. Greubel, Aug 29 2018

Crossrefs

Cf. A001008, A001620, A131915, A131917, A131918.

Sequence in context: A197477 A133839 A080828 * A131606 A033328 A097529

Adjacent sequences: A131913 A131914 A131915 * A131917 A131918 A131919

Keyword

cofr,easy,nonn

Author

Jonathan Vos Post, Jul 27 2007

Status

approved