A222362 -id:A222362 - cp's OEIS frontend

A103710 Decimal expansion of the ratio of the length of the latus rectum arc of any parabola to its semi latus rectum: sqrt(2) + log(1 + sqrt(2)).

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

Offset: 1

Sylvester Reese and Jonathan Sondow, Feb 13 2005

The universal parabolic constant, equal to the ratio of the latus rectum arc of any parabola to its focal parameter. Like Pi, it is transcendental.
Just as all circles are similar, all parabolas are similar. Just as the ratio of a semicircle to its radius is always Pi, the ratio of the latus rectum arc of any parabola to its semi latus rectum is sqrt(2) + log(1 + sqrt(2)).
Note the remarkable similarity to sqrt(2) - log(1 + sqrt(2)), the universal equilateral hyperbolic constant A222362, which is a ratio of areas rather than of arc lengths. Lockhart (2012) says "the arc length integral for the parabola .. is intimately connected to the hyperbolic area integral ... I think it is surprising and wonderful that the length of one conic section is related to the area of another."
Is it a coincidence that the universal parabolic constant is equal to 6 times the expected distance A103712 from a randomly selected point in the unit square to its center? (Reese, 2004; Finch, 2012)

			2.29558714939263807403429804918949038759783220363858348392997534664...

H. Dörrie, 100 Great Problems of Elementary Mathematics, Dover, 1965, Problems 57 and 58.
P. Lockhart, Measurement, Harvard University Press, 2012, p. 369.
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.

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).
S. R. Finch, Mathematical Constants, Errata and Addenda, 2012, 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, Jonathan Sondow, Eric W. Weisstein, MathWorld: Universal Parabolic Constant
Manas Shetty, Prajwal DSouza, Sparsha Kumari, Vinton Adrian Rebello, Where is the Parabola? The Parabolic Constant Mystery
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.
Wikipedia, Universal parabolic constant
Index entries for transcendental numbers

A002193 + A091648.

Cf. A103711, A103712, A222362, A232716, A232717.

RealDigits[ Sqrt[2] + Log[1 + Sqrt[2]], 10, 111][[1]] (* Robert G. Wilson v Feb 14 2005 *)

fpprec: 100$ ev(bfloat(sqrt(2) + log(1 + sqrt(2)))); /* Martin Ettl, Oct 17 2012 */

sqrt(2)+log(1+sqrt(2)) \\ Charles R Greathouse IV, Mar 08 2013

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

A180434 Decimal expansion of constant (2 - Pi/2).

Original entry on oeis.org

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

Offset: 0

Jonathan Vos Post, Sep 05 2010

(2-Pi/2)*a^2 is the area of the loop of the right strophoid (also called the Newton strophoid) whose polar equation is r = a*cos(2*t)/cos(t) and whose Cartesian equation is x*(x^2+y^2) = a*(x^2-y^2) or y = +- x*sqrt((a-x)/(a+x)). See the curve with its loop at the Mathcurve link; the loop appears for -Pi/4 <= t <= Pi/4. - Bernard Schott, Jan 28 2020

			0.42920367320510338076867830836024855790141530...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Robert Ferréol, Right strophoid, Math Curve.
Nikita Kalinin and Mikhail Shkolnikov, The number Pi and summation by SL(2,Z), arXiv:1701.07584 [math.NT], 2016. Gives a formula.
Index entries for transcendental numbers

Cf. A004601, A153799, A180433, A222362.

RealDigits[2-Pi/2,10,120][[1]] (* Harvey P. Dale, Oct 12 2013 *)

Equals Integral_{t=0..Pi/4} ((cos(2*t))/cos(t))^2 dt. - Bernard Schott, Jan 28 2020
From Amiram Eldar, May 30 2021: (Start)
Equals Sum_{k>=1} 2^k/(binomial(2*k,k)*k*(2*k + 1)).
Equals Integral_{x=0..1} arcsin(x)*arccos(x) dx. (End)
Equals Integral_{x=0..1} sqrt(x)/(1+x) dx. - Andy Nicol, Mar 23 2024
Equals A153799/2. - Hugo Pfoertner, Mar 23 2024

Corrected by Carl R. White, Sep 09 2010

More terms from N. J. A. Sloane, Sep 23 2010

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.

Original entry on oeis.org

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

Sylvester Reese and Jonathan Sondow, Feb 13 2005

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)

			1.14779357469631903701714902459474519379891610181929174196498767332...

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.

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

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

Cf. A103711, A222362, A232716, A232717.

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 *)

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

A278386 Decimal expansion of the excess of the exponential curve arc length over the length of the x-axis from -infinity to zero.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Nov 20 2016

			0.22598715591349733298631152068808233761701168147556791654406413883...

Jean-François Alcover, Involute of the exponential curve (left branch).

Cf. A222362 (a similar constant).

RealDigits[Sqrt[2] - 1 + Log[2] - Log[1 + Sqrt[2]], 10, 100][[1]]
RealDigits[Sqrt[2] - 1 - ArcSinh[7/(4 (5 + 3 Sqrt[2]))], 10, 100][[1]] (* Jan Mangaldan, Nov 22 2020 *)

sqrt(2) - 1 + log(2) - log(1 + sqrt(2)) \\ Michel Marcus, Nov 20 2016

Equals Integral_{-infinity..0} (sqrt(1 + exp(2x))-1) dx.
Equals sqrt(2) - 1 + log(2) - log(1 + sqrt(2)).
Equals sqrt(2) - 1 - arcsinh(7/(4*(5 + 3*sqrt(2)))). - Jan Mangaldan, Nov 23 2020
Equals sqrt(2) - 1 - arcsinh((5 - 3*sqrt(2))/4). - Vaclav Kotesovec, Nov 27 2020
Equals Integral_{x=0..1} (sqrt(x^2 + 1) - 1)/x dx. - Kritsada Moomuang, May 27 2025

cp's OEIS Frontend

A103710 Decimal expansion of the ratio of the length of the latus rectum arc of any parabola to its semi latus rectum: sqrt(2) + log(1 + sqrt(2)).

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A180434 Decimal expansion of constant (2 - Pi/2).

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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.

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A278386 Decimal expansion of the excess of the exponential curve arc length over the length of the x-axis from -infinity to zero.

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