cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 13 results. Next

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

Original entry on oeis.org

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

Views

Author

Eric W. Weisstein, Jan 24 2004

Keywords

Comments

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

Examples

			0.8813735870195430252326093249797923090281603282616...
		

References

  • 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.

Crossrefs

Programs

  • Mathematica
    RealDigits[Log[1 + Sqrt[2]], 10, 100][[1]] (* Alonso del Arte, Aug 11 2011 *)
  • Maxima
    fpprec : 100$ ev(bfloat(log(1 + sqrt(2)))); /* Martin Ettl, Oct 17 2012 */
    
  • PARI
    asinh(1) \\ Michel Marcus, Oct 19 2014

Formula

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

Views

Author

Jean-François Alcover, Jul 08 2014

Keywords

Comments

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

Examples

			0.76519571646421269134476601639649679586594406787952782797665844888136988756...
		

Crossrefs

Cf. A244920.

Programs

  • Mathematica
    RealDigits[(Sqrt[2] + Log[1 + Sqrt[2]])/3, 10, 104] // First
  • PARI
    (sqrt(2)+log(1+sqrt(2)))/3 \\ G. C. Greubel, Jul 05 2017

Formula

Also equals (sqrt(2) + arcsinh(1))/3.
This is also 2*A103712. - Derek Orr, Jul 27 2014

A254979 Decimal expansion of the mean Euclidean distance from a point in a unit 4D cube to a given vertex of the cube (named B_4(1) in Bailey's paper).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Feb 11 2015

Keywords

Comments

Also, decimal expansion of twice the expected distance from a randomly selected point in the unit 4D cube to the center. - Amiram Eldar, Jun 04 2023

Examples

			1.12189961871586097735161517556754270920080795643954583...
		

Crossrefs

Analogous constants: A244921 (square), A130590 (cube).

Programs

  • Mathematica
    Ti2[x_] := (I/2)*(PolyLog[2, -I*x] - PolyLog[2, I*x]); B4[1] = 2/5 - Catalan/10 + (3/10)*Ti2[3 - 2*Sqrt[2]] + Log[3] - (7*Sqrt[2]/10)*ArcTan[1/Sqrt[8]] // Re; RealDigits[B4[1], 10, 105] // First
    N[Integrate[1/u^2 - Pi^2*Erf[u]^4/(16*u^6), {u, 0, Infinity}]/Sqrt[Pi], 50] (* Vaclav Kotesovec, Aug 13 2019 *)
  • Python
    from mpmath import *
    mp.dps=106
    x=3 - 2*sqrt(2)
    Ti2x=(j/2)*(polylog(2, -j*x) - polylog(2, j*x))
    C = 2/5 - catalan/10 + (3/10)*Ti2x + log(3) - (7*sqrt(2)/10)*atan(1/sqrt(8))
    print([int(n) for n in str(C.real).replace('.', '')]) # Indranil Ghosh, Jul 04 2017

Formula

Equals B_4(1) = 2/5 - Catalan/10 + (3/10)*Ti_2(3-2*sqrt(2)) + log(3) - (7*sqrt(2)/10) * arctan(1/sqrt(8)), where Ti_2(x) = (i/2)*(polylog(2, -i*x) - polylog(2, i*x)) (Ti_2 is the inverse tangent integral function).

Extensions

Name corrected by Amiram Eldar, Jun 04 2023

A244922 Decimal expansion of the integral over the square [0,1]x[0,1] of (x^2 + y^2)^(3/2) dx dy.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jul 08 2014

Keywords

Examples

			0.627180784883514720865482452220363173853609205621177137224832249594762945...
		

Crossrefs

Programs

  • Mathematica
    RealDigits[7/20*Sqrt[2] + 3/20*Log[1 + Sqrt[2]], 10, 106] // First

Formula

7/20*sqrt(2) + 3/20*log(1 + sqrt(2)).
Also equals (7*sqrt(2) + 3*arcsinh(1))/20.

A322197 Antidiagonal sums of square table A322190.

Original entry on oeis.org

1, 2, 3, 6, 15, 46, 168, 710, 3405, 18270, 108438, 705334, 4989075, 38126414, 313034088, 2748039078, 25685633625, 254672239678, 2669718010218, 29502715813142, 342784073066655, 4177349457737262, 53279132429530428, 709785147883342726, 9858698782067445765, 142530638751865262366, 2141519206261256136318, 33391802751245681847030, 539616796036523449056555, 9026558167976152019922190
Offset: 0

Views

Author

Paul D. Hanna, Dec 20 2018

Keywords

Comments

Table A322190 gives the coefficients of x^n*y^k/(n!*k!) in (cosh(x)*cosh(y) + sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)).

Crossrefs

Cf. A244920.

Programs

  • Mathematica
    nmax = 30;
    t[n_, k_] := SeriesCoefficient[(Cosh[x] Cosh[y] + Sinh[x] + Sinh[y])/(1 - Sinh[x] Sinh[y]), {x, 0, n}, {y, 0, k}] n! k!;
    a[n_] := Sum[t[n - k, k], {k, 0, n}];
    Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Dec 29 2018 *)

Formula

a(n) ~ Pi * n^(n+1) / (2^(n - 3/4) * exp(n) * (log(1+sqrt(2)))^(n + 3/2)). - Vaclav Kotesovec, Dec 30 2018

A254133 Decimal expansion of Lamb's integral K_0.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jan 26 2015

Keywords

Examples

			0.490772728958345159162717253203382640381923347758584656...
		

Crossrefs

Programs

  • Maple
    evalf(int(arctanh(1/sqrt(3 + x^2))/(1 + x^2), x=0..1), 120); # Vaclav Kotesovec, Jan 26 2015
  • Mathematica
    Ti2[x_] := (I/2)*(PolyLog[2, -I*x] - PolyLog[2, I*x]); K0 = (3/2)*Ti2[3 - 2 Sqrt[2]] + Pi/4*Log[1 + Sqrt[2]] - Catalan/2 // Re; RealDigits[K0, 10, 103] // First

Formula

K_0 = integral_[0..1] arctanh(1/sqrt(3 + x^2))/(1 + x^2) dx.
K_0 = 3/2*Ti_2(3 - 2*sqrt(2)) + Pi/4*log(1 + sqrt(2)) - Catalan/2, where Ti_2 is Lewin's arctan integral, Ti_2(x) = (i/2)*(Li_2(-i*x) - Li_2(i*x)).

A254134 Decimal expansion of Lamb's integral K_1.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jan 26 2015

Keywords

Examples

			1.6619078747381233774065816861630594973488686732512589...
		

Crossrefs

Programs

  • Maple
    evalf(int(arcsec(x)/sqrt(x^2 - 4*x + 3), x=3..4), 120); # Vaclav Kotesovec, Jan 26 2015
  • Mathematica
    Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]); th = (ArcTan[(16 - 3*Sqrt[15])/11] + Pi)/3; K1 = Cl2[th] - Cl2[th + Pi/3] - Cl2[th - Pi/2] + Cl2[th - Pi/6] - Cl2[3*th + Pi/3] + Cl2[3*th + 2*(Pi/3)] - Cl2[3*th - 5*(Pi/6)] + Cl2[3*th + 5*(Pi/6)] + (6*th - 5*(Pi/2))*Log[2 - Sqrt[3]] // Re; RealDigits[K1, 10, 103] // First

Formula

K_1 = integral_[3..4] arcsec(x)/sqrt(x^2 - 4*x + 3) dx.
K_1 = Cl_2(th) - Cl_2(th + Pi/3) - Cl_2(th - Pi/2) + Cl_2(th - Pi/6) - Cl_2(3*th + Pi/3) + Cl_2(3*th + 2*(Pi/3)) - Cl_2(3*th - 5*(Pi/6)) + Cl_2(3*th + 5*(Pi/6)) + (6*th - 5*(Pi/2))*log(2 - sqrt(3)), where Cl_2 is the Clausen function and th = (arctan((16 - 3*sqrt(15))/11) + Pi)/3.

A254135 Decimal expansion of Lamb's integral K_2.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jan 26 2015

Keywords

Examples

			0.69266081515264750650943118588427245846713483280766884258...
		

Crossrefs

Programs

  • Maple
    evalf(int(sqrt(1 + sec(x)^2)*arctan(1/sqrt(1 + sec(x)^2)), x=0..Pi/4), 120); # Vaclav Kotesovec, Jan 26 2015
  • Mathematica
    Ti2[x_] := (I/2)* (PolyLog[2, -I *x] - PolyLog[2, I *x]); K2 = (1/2)*Ti2[-2 + Sqrt[3]] + (Pi/8)*Log[2 + Sqrt[3]] + Pi^2/32 // Re; RealDigits[K2, 10, 103] // First

Formula

K_2 = integral_[0..Pi/4] sqrt(1 + sec(x)^2)*arctan(1/sqrt(1 + sec(x)^2)) dx.
K_2 = (1/2)*Ti_2(-2 + sqrt(3)) + (Pi/8)*log(2 + sqrt(3)) + Pi^2/32, where Ti_2 is Lewin's arctan integral, Ti_2(x) = (i/2)*(Li_2(-i*x) - Li_2(i*x)).

A254968 Decimal expansion of the mean reciprocal Euclidean distance from a point in a unit cube to a given vertex of the cube (named B_3(-1) in Bailey's paper).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Feb 11 2015

Keywords

Examples

			1.1900386819897767533219086751420769449911806073574982644...
		

Crossrefs

Programs

  • Mathematica
    RealDigits[(3/2)*Log[2 + Sqrt[3]] - Pi/4, 10, 105] // First

Formula

Equals B_3(-1) = (3/2)*log(2 + sqrt(3)) - Pi/4.
Equals log(7 + 4*sqrt(3)) - Pi/4 - arcsinh(1/sqrt(2)).

Extensions

Name corrected by Amiram Eldar, Jun 04 2023

A377144 Decimal expansion of Integral_{x=0..oo} sin(x^2)*erfc(x) dx, where erfc is the complementary error function.

Original entry on oeis.org

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

Views

Author

Paolo Xausa, Oct 17 2024

Keywords

Examples

			0.137519939983291533628377694180502292463433867842...
		

Crossrefs

Programs

  • Mathematica
    First[RealDigits[(Pi - 2*ArcSinh[1])/(4*Sqrt[2*Pi]), 10, 100]]

Formula

Equals (Pi - 2*arcsinh(1))/(4*sqrt(2*Pi)) = (A000796 - A244920)/(4*A019727) (cf. eq. 12 in Weisstein link).
Showing 1-10 of 13 results. Next