A103711 Decimal expansion of the ratio of the length of the latus rectum arc of any parabola to its latus rectum: (sqrt(2) + log(1 + sqrt(2)))/2.

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

Offset: 1

Author

Sylvester Reese and Jonathan Sondow, Feb 13 2005

Keywords

• cons
• easy
• nonn

Comments

Just as all circles are similar, all parabolas are similar. Just as the ratio of a semicircle to its diameter is always Pi/2, the ratio of the length of the latus rectum arc of any parabola to its latus rectum is (sqrt(2) + log(1 + sqrt(2)))/2.

Let c = this constant and a = e - exp((c+Pi)/2 - log(Pi)), then a = .0000999540234051652627... and c - 10*(-log(exp(a) - a - 1) - 19) = .000650078964115564700067717... - Gerald McGarvey, Feb 21 2005

Half the universal parabolic constant A103710 (the ratio of the length of the latus rectum arc of any parabola to its focal parameter). Like Pi, it is transcendental.

Is it a coincidence that this constant is equal to 3 times the expected distance A103712 from a randomly selected point in the unit square to its center? (Reese, 2004; Finch, 2012)

Examples

			1.14779357469631903701714902459474519379891610181929174196498767332...
		

References

• H. Dörrie, 100 Great Problems of Elementary Mathematics, Dover, 1965, Problems 57 and 58.
• C. E. Love, Differential and Integral Calculus, 4th ed., Macmillan, 1950, pp. 286-288.
• C. S. Ogilvy, Excursions in Geometry, Oxford Univ. Press, 1969, p. 84.
• S. Reese, A universal parabolic constant, 2004, preprint.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• J. L. Diaz-Barrero and W. Seaman, A limit computed by integration, Problem 810 and Solution, College Math. J., 37 (2006), 316-318, equation (5).
• Steven R. Finch, Mathematical Constants, Errata and Addenda, arXiv:2001.00578 [math.HO], 2012-2024, section 8.1.
• M. Hajja, Review Zbl 1291.51018, zbMATH 2015.
• M. Hajja, Review Zbl 1291.51016, zbMATH 2015.
• H. Khelif, L’arbelos, Partie II, Généralisations de l’arbelos, Images des Mathématiques, CNRS, 2014.
• J. Pahikkala, Arc Length Of Parabola, PlanetMath.
• S. Reese, Pohle Colloquium Video Lecture: The universal parabolic constant, Feb 02 2005
• S. Reese and Jonathan Sondow, Universal Parabolic Constant, MathWorld
• Jonathan 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 2012, Amer. Math. Monthly, 121 (2014), 438-443.
• Eric Weisstein's World of Mathematics, Universal Parabolic Constant
• Wikipedia, Universal parabolic constant
• Index entries for transcendental numbers

Crossrefs

Equal to (A103710)/2 = (A002193 + A091648)/2 = 3*(A103712).

Cf. A103711, A222362, A232716, A232717.

Programs

Mathematica

RealDigits[(Sqrt[2] + Log[1 + Sqrt[2]])/2, 10, 111][[1]] (* Robert G. Wilson v, Feb 14 2005 *)
N[Integrate[Sqrt[1 + x^2], {x, 0, 1}], 120] (* Clark Kimberling, Jan 06 2014 *)

Formula

Equals Integral_{x = 0..1} sqrt(1 + x^2) dx. - Peter Bala, Feb 28 2019

Equals Sum_{n>=0} (-1)^(n + 1)*binomial(2*n, n)/((4*n^2 - 1)*4^n). - Antonio Graciá Llorente, Dec 16 2024