A232716 Decimal expansion of the ratio of the length of the boundary of any parbelos to the length of the boundary of its associated arbelos: (sqrt(2) + log(1 + sqrt(2))) / Pi.

7, 3, 0, 7, 0, 8, 0, 8, 4, 2, 4, 8, 1, 4, 3, 0, 9, 8, 3, 4, 5, 4, 5, 9, 3, 8, 9, 9, 7, 0, 9, 9, 0, 1, 3, 7, 7, 3, 6, 7, 2, 3, 2, 8, 7, 2, 9, 1, 6, 6, 0, 2, 7, 5, 7, 3, 5, 4, 9, 8, 3, 9, 1, 9, 5, 1, 0, 0, 7, 2, 9, 3, 2, 5, 3, 5, 5, 1, 3, 5, 4, 0, 2, 6, 0, 1, 4, 0, 8, 2, 9, 3, 5, 0, 7, 6, 2, 1, 1, 9, 6

Offset: 0

Author

Jonathan Sondow, Nov 28 2013

Keywords

• nonn
• cons
• easy

Comments

Same as decimal expansion of P/Pi, where P is the Universal parabolic constant (A103710). - Jonathan Sondow, Jan 19 2015

According to Wadim Zudilin, Campbell's formula (see below) follows from results of Borwein, Borwein, Glasser, Wan (2011): Take n=-2, s=1/4 in equations (4) and (20) to see that the formula is about evaluating K_{-2,1/4}. Take r=-1/2, s=1/4 in (76) to see that K_{-2,1/4} = cos(Pi/4)-K_{0,1/4}/16. Finally, use (51) and (52) to conclude that K_{0,1/4} = 2G_{1/4} = 2*log(1+sqrt(2)). - Jonathan Sondow, Sep 03 2016

Examples

			0.730708084248143098345459389970990137736723287291660275735498...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• D. Borwein, J. M. Borwein, M. L. Glasser, J. G. Wan, Moments of Ramanujan's generalized elliptic integrals and extensions of Catalan's constant, J. Math. Anal. Appl., 384 (2) (2011), 478-496.
• M. Hajja, Review Zbl 1291.51018, zbMATH 2015.
• M. Hajja, Review Zbl 1291.51016, zbMATH 2015.
• J. Sondow, The parbelos, a parabolic analog of the arbelos, arXiv 2012, Amer. Math. Monthly, 120 (2013), 929-935.
• E. Tsukerman, Solution of Sondow's problem: a synthetic proof of the tangency property of the parbelos, arXiv:1210.5580 [math.MG], 2012-2013; Amer. Math. Monthly, 121 (2014), 438-443.

Crossrefs

Reciprocal of A232717. Ratio of areas is A177870.

Cf. A000796, A103710.

Programs

Magma

R:= RealField(); (Sqrt(2) + Log(1 + Sqrt(2)))/Pi(R); // G. C. Greubel, Feb 02 2018

Mathematica

RealDigits[(Sqrt[2] + Log[1 + Sqrt[2]])/Pi,10,100]

PARI

(sqrt(2) + log(1 + sqrt(2)))/Pi \\ G. C. Greubel, Feb 02 2018

Formula

Equals A103710 / A000796.

Empirical: equals 3F2([-1/2,1/4,3/4],[1/2,1],1). - John M. Campbell, Aug 27 2016