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

0,1

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

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.

Formula

Equals A103710 / A000796.

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

Example

0.730708084248143098345459389970990137736723287291660275735498...

Mathematica

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

Prog

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

Crossrefs

Reciprocal of A232717. Ratio of areas is A177870.

Cf. A000796, A103710.

Sequence in context: A171535 A245082 A019648 * A327574 A253905 A154159

Adjacent sequences: A232713 A232714 A232715 * A232717 A232718 A232719

Keyword

nonn,cons,easy

Author

Jonathan Sondow, Nov 28 2013

Status

approved