A131916 - OEIS

%I #28 Aug 10 2024 16:23:10

%S 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,

%T 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,

%U 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

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

%C Decimal expansion is A131915.

%H G. C. Greubel, <a href="/A131916/b131916.txt">Table of n, a(n) for n = 0..9999</a>

%H Mark B. Villarino, <a href="http://arXiv.org/abs/0707.3950">Ramanujan's Harmonic Number Expansion into Negative Powers of a Triangular Number</a>, arXiv:0707.3950 [math.CA], 2007. Constant occurs in Theorem 6, formula (1.13), page 6.

%F _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.

%e 1.1215093407... = 1 + 1/(8 + 1/(4 + 1/(2 + 1/(1 + 1/(5 + ...))))).

%t ContinuedFraction[1/(1-EulerGamma-Log[Sqrt[2]])-12,100] (* _Harvey P. Dale_, Oct 01 2015 *)

%o (PARI) contfrac(1/(1 - Euler - log(sqrt(2))) - 12) \\ _Michel Marcus_, Mar 11 2013

%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); ContinuedFraction(1/(1 - EulerGamma(R) - Log(Sqrt(2))) - 12); // _G. C. Greubel_, Aug 29 2018

%Y Cf. A001008, A001620, A131915 (decimal expansion), A131917, A131918.

%K cofr,easy,nonn

%O 0,2

%A _Jonathan Vos Post_, Jul 27 2007

%E Offset changed by _Andrew Howroyd_, Aug 10 2024