A103712 -id:A103712 - cp's OEIS frontend

A091648 Decimal expansion of arccosh(sqrt(2)), the inflection point of sech(x).

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

Offset: 0

Eric W. Weisstein, Jan 24 2004

Asymptotic growth constant in the exponent for the number of spanning trees on the 2 X infinity strip on the square lattice. - R. J. Mathar, May 14 2006
Arccosh(sqrt(2)) = (1/2)*log((sqrt(2)+1)/(sqrt(2)-1)) = log(tan(3*Pi/8)) = int(1/cos(x),x=0..Pi/4). Therefore, in Gerardus Mercator's (conformal) map this is the value of the ordinate y/R (R radius of the spherical earth) for latitude phi = 45 degrees north, or Pi/4. See, e.g., the Eli Maor reference, eqs. (5) and (6). This is the latitude of, e.g., the Mission Point Lighthouse, Michigan, U.S.A. - Wolfdieter Lang, Mar 05 2013
Decimal expansion of the arclength on the hyperbola y^2 - x^2 = 1 from (0,0) to (1,sqrt(2)). - Clark Kimberling, Jul 04 2020

			0.8813735870195430252326093249797923090281603282616...

L. B. W. Jolley, Summation of Series, Dover (1961), Eq. (85) page 16-17.
E. Maor, Trigonometric Delights, Princeton University Press, NJ, 1998, chapter 13, A Mapmaker's Paradise, pp. 163-180.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 30, equation 30:10:4 at page 283.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
E. D. Krupnikov, K. S. Kölbig, Some special cases of the generalized hypergeometric function (q+1)Fq, J. Comp. Appl. Math. 78 (1997) 79-95.
D. H. Lehmer, Interesting Series Involving the Central Binomial Coefficient, Am. Math. Monthly 92 (1985) 449.
Michael I. Shamos, A catalog of the real numbers, (2007). See pp. 614-615.
R. Shrock and F. Y. Wu, Spanning trees on graphs and lattices in d dimensions, J Phys A: Math Gen 33 (2000) 3881-3902.
Eric Weisstein's World of Mathematics, Hyperbolic Secant.
Eric Weisstein's World of Mathematics, Universal Parabolic Constant.
Index entries for transcendental numbers.

Cf. A014176, A103710, A103711, A103712, A181048.

RealDigits[Log[1 + Sqrt[2]], 10, 100][[1]] (* Alonso del Arte, Aug 11 2011 *)

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

PARI
```
asinh(1) \\ Michel Marcus, Oct 19 2014
```

Equals log(1 + sqrt(2)). - Jonathan Sondow, Mar 15 2005
Equals (1/2)*log(3+2*sqrt(2)) = A244920/2. - R. J. Mathar, May 14 2006
Equals Sum_{n>=1, n odd} binomial(2*n,n)/(n*4^n) [see Lehmer link]. - R. J. Mathar, Mar 04 2009
Equals arcsinh(1), since arcsinh(x) = log(x+sqrt(x^2+1)). - Stanislav Sykora, Nov 01 2013
Equals asin(i)/i. - L. Edson Jeffery, Oct 19 2014
Equals (Pi/4) * 3F2(1/4, 1/2, 3/4; 1, 3/2; 1). - Jean-François Alcover, Apr 23 2015
Equals arctanh(sqrt(2)/2). - Amiram Eldar, Apr 22 2022
Equals lim_{n->oo} Sum_{k=1..n} 1/sqrt(n^2+k^2). - Amiram Eldar, May 19 2022
Equals Sum_{n >= 1} 1/(n*P(n, sqrt(2))*P(n-1, sqrt(2))), where P(n, x) denotes the n-th Legendre polynomial. The first twenty terms of the series gives the approximate value 0.88137358701954(24...), correct to 14 decimal places. - Peter Bala, Mar 16 2024
Equals 2F1(1/2,1/2;3/2;-1) [Krupnikov]. - R. J. Mathar, May 13 2024
Equals Integral_{x=0..1} (x^sqrt(2) - 1)/log(x) dx. - Kritsada Moomuang, Jun 06 2025

A244921 Decimal expansion of (sqrt(2)+log(1+sqrt(2)))/3, the integral over the square [0,1]x[0,1] of sqrt(x^2+y^2) dx dy.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jul 08 2014

This is also the expected distance from a randomly selected point in the unit square to a corner, as well as the expected distance from a randomly selected point in a 45-45-90 degree triangle of base length 1 to a vertex with an acute angle. - Derek Orr, Jul 27 2014

The average length of chords in a unit square drawn between two points uniformly and independently chosen at random on two adjacent sides. - Amiram Eldar, Aug 08 2020

			0.76519571646421269134476601639649679586594406787952782797665844888136988756...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. Bailey, J. Borwein, and R. Crandall, Advances in the theory of box integrals, Mathematics of Computation, Vol. 79, No. 271 (2010), pp. 1839-1866. See p. 1860.
Philip W. Kuchel and Rodney J. Vaughan, Average lengths of chords in a square, Mathematics Magazine, Vol. 54, No. 5 (1981), pp. 261-269.
Index entries for transcendental numbers

Cf. A244920.

RealDigits[(Sqrt[2] + Log[1 + Sqrt[2]])/3, 10, 104] // First

(sqrt(2)+log(1+sqrt(2)))/3 \\ G. C. Greubel, Jul 05 2017

Also equals (sqrt(2) + arcsinh(1))/3.

This is also 2*A103712. - Derek Orr, Jul 27 2014

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)).

Original entry on oeis.org

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

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

A355183 Decimal expansion of the area of the region that represents the set of points in a unit square that are closer to the center of the square than to the closest edge.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 23 2022

The shape is formed by the intersection of four parabolas. Its perimeter is given in A355184.

			0.21895141649746006506891829894626410475956250050259...

Kiran S. Kedlaya, Bjorn Poonen, and Ravi Vakil, The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions, and Commentary, The Mathematical Association of America, 2002, pp. 108-109.

Joel Atkins, Regular Polygon Targets, Pi Mu Epsilon Journal, Vol. 9, No. 3 (1989), pp. 142-144; entire issue.
Nicholas R. Baeth, Loren Luther, and Rhonda McKee, The Downtown Problem: Variations on a Putnam Problem, Mathematics Magazine, Vol. 90, No. 4 (2017), pp. 243-257.
John Coffey, Q19, Maths Puzzles & Problems, MathStudio, 2011.
Amiram Eldar, Illustration.
Leonard F. Klosinski, Gerald L. Alexanderson, and Loren C. Larson, The Fiftieth William Lowell Putnam Mathematical Competition, The American Mathematical Monthly, Vol. 98, No. 4 (1991), pp. 319-327.
Missouri State University, Problem #5, The Area and Perimeter of a Certain Region, Advanced Problem Archive; Solution to Problem #5, by John Shonder.
Jun-Ping Shi, Problem Set 9.

Cf. A021058, A103712, A244921, A254140, A352453, A355184 (perimeter), A355185 (3D analog).

RealDigits[(4*Sqrt[2] - 5)/3, 10, 100][[1]]

Equals (4*sqrt(2)-5)/3.

A345653 Decimal expansion of (sqrt(2) + arcsinh(1))/4.

Original entry on oeis.org

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

Offset: 0

Thomas Anton, Jun 21 2021

Maximal value of the average distance from the set of lattice points of any line in the plane, where average distance is the limit of the integral of the distance to the set over a segment of the line centered around a fixed point, divided by the length of the segment, as that length approaches infinity, achieved precisely by lines x = z + 1/2 and y = z + 1/2 for integers z.

The average distance between the center of a unit square to a randomly and uniformly chosen point on its perimeter. - Amiram Eldar, Jun 23 2022

			0.57389678734815951850857451...

Thomas Anton, Table of n, a(n) for n = 0...259
Index entries for transcendental numbers

Cf. A000328, A103710, A103711, A103712.

First[RealDigits[N[(Sqrt[2]+ArcSinh[1])/4,100]]] (* Stefano Spezia, Jun 21 2021 *)

(sqrt(2) + asinh(1))/4 \\ Michel Marcus, Jun 24 2021

Equals Integral_{x=0..1/2} 2*sqrt(x^2+1/4) dx.

Equals (1/2) * A103711. - Amiram Eldar, Jun 23 2022

A245698 Decimal expansion of the expected distance from a randomly selected point in an equilateral triangle of side length 1 to its center: (2*sqrt(3) + log(2+sqrt(3)))/18.

Original entry on oeis.org

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

Offset: 1

Derek Orr, Jul 29 2014

			0.265614417336809516426663279462206287661810693282682096437...

Eric Weisstein's World of Mathematics, Triangle Point Picking
Index entries for transcendental numbers

Cf. A103712.

RealDigits[(2Sqrt[3]+Log[2+Sqrt[3]])/18,10,120][[1]] (* Harvey P. Dale, Aug 09 2014 *)

sqrt(3)/9 + log(sqrt(3)+2)/18 \\ Charles R Greathouse IV, May 15 2019

Also equal to (8*sqrt(3)+3*arcsinh(sqrt(3))+log(2+sqrt(3)))/72.

A245699 Decimal expansion of the expected distance from a randomly selected point in a 45-45-90 degree triangle of base length 1 to the vertex of the right angle: (4+sqrt(2)log(3+2sqrt(2)))/12.

Original entry on oeis.org

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

Offset: 0

Derek Orr, Jul 29 2014

			0.54107508004674350446467336008352266755023177078218908429957159203205...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers

Cf. A103712.

SetDefaultRealField(RealField(100)); (4+Sqrt(2)*Log(3 +2*Sqrt(2)))/12; // G. C. Greubel, Oct 06 2018

evalf((4+sqrt(2)*log(3+2*sqrt(2)))/12,100); # Muniru A Asiru, Oct 07 2018

RealDigits[(4 + Sqrt[2]*Log[3 + 2*Sqrt[2]])/12, 10, 100][[1]] (* G. C. Greubel, Oct 06 2018 *)

default(realprecision, 100); (4+sqrt(2)*log(3+2*sqrt(2)))/12 \\ G. C. Greubel, Oct 06 2018

Equals Integral_{y = 0..Pi/4; x = 0..1/(sqrt(2)*cos(y))} 4x^2 dx dy.

Equals Integral_{y = 0..Pi/4} (sqrt(2)/3)*sec^3(y) dy.

A355184 Decimal expansion of the perimeter of the region that represents the set of points in a unit square that are closer to the center of the square than to the closest edge.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 23 2022

The shape is formed by the intersection of four parabolas. Its area is given in A355183.

			1.70308249665895322783584912274920315719803442295049...

Amiram Eldar, Illustration.
Missouri State University, Problem #5, The Area and Perimeter of a Certain Region, Advanced Problem Archive; Solution to Problem #5, by John Shonder.

Cf. A103712, A244921, A254140, A352453, A355183 (area).

RealDigits[2*Log[Sqrt[4 - 2*Sqrt[2]] + Sqrt[2] - 1] - Sqrt[16 - 8*Sqrt[2]] + Sqrt[32 - 16*Sqrt[2]], 10, 100][[1]]

Equals 2*log(sqrt(4-2*sqrt(2))+sqrt(2)-1) - sqrt(16-8*sqrt(2)) + sqrt(32-16*sqrt(2)).

cp's OEIS Frontend

A091648 Decimal expansion of arccosh(sqrt(2)), the inflection point of sech(x).

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A244921 Decimal expansion of (sqrt(2)+log(1+sqrt(2)))/3, the integral over the square [0,1]x[0,1] of sqrt(x^2+y^2) dx dy.

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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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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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A355183 Decimal expansion of the area of the region that represents the set of points in a unit square that are closer to the center of the square than to the closest edge.

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A345653 Decimal expansion of (sqrt(2) + arcsinh(1))/4.

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A245698 Decimal expansion of the expected distance from a randomly selected point in an equilateral triangle of side length 1 to its center: (2*sqrt(3) + log(2+sqrt(3)))/18.

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A245699 Decimal expansion of the expected distance from a randomly selected point in a 45-45-90 degree triangle of base length 1 to the vertex of the right angle: (4+sqrt(2)log(3+2sqrt(2)))/12.