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 1-5 of 5 results.

A197682 Decimal expansion of Pi/(2 + 2*Pi).

Original entry on oeis.org

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

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Author

Clark Kimberling, Oct 17 2011

Keywords

Comments

The number Pi/(2 + 2*Pi) is the least x > 0 such that sin(x) = cos(Pi*x).
If b and c are distinct real numbers, the solutions of sin(bx) = cos(cx) are x = (k - 1/2)*Pi/(b + c), where k runs through the integers. Thus, if b > 0 and c > 0, the least solution x > 0 is Pi/(2*b + 2*c), so that this is also the least x > 0 for which sin(c*x) = cos(b*x). Related sequences, each with a Mathematica program which includes a graph:
...
b.....c.......sequence........x
1.....2.......A019673........ x = Pi/6
1.....3.......A019678........ x = Pi/8
1.....4.......(A000796)/10... x = Pi/10
1.....Pi......A197682........ x = Pi/(2+2*Pi)
1.....2*Pi....A197683........ x = Pi/(2+4*Pi)
1.....1/Pi....A197684........ x = Pi^2/(2+2*Pi)
1.....2/Pi....A197685........ x = Pi^2/(4+2*Pi)
1.....Pi/2....A197686........ x = Pi/(2+Pi)
1.....Pi/3....A197687........ x = 3*Pi/(6+2*Pi)
1.....Pi/4....A197688........ x = 2*Pi/(4+Pi)
1.....Pi/6....A197689........ x = 3*Pi/(6+Pi)
2.....3.......(A000796)/10... x = Pi/10
2.....Pi......A197690........ x = Pi/(4+2*Pi)
2.....2*Pi....A197691........ x = Pi/(4+4*Pi)
2.....1/Pi....A197692........ x = Pi^2/(2+4*Pi)
2.....2/Pi....A197693........ x = Pi^2/(4+4*Pi)
2.....Pi/2....A197694........ x = Pi/(4+Pi)
3.....Pi......A197695........ x = Pi/(2+2*Pi)
3.....2*Pi....A197696........ x = Pi/(6+4*Pi)
3.....1/Pi....A197697........ x = Pi^2/(2+6*Pi)
3.....2/Pi....A197698........ x = Pi^2/(4+6*Pi)
3.....Pi/2....A197699........ x = Pi/(6+Pi)
1/2...Pi......A197700........ x = Pi/(1+2*Pi)
1/2...2*Pi....A197701........ x = Pi/(1+4*Pi)
1/2...1/Pi....A197724........ x = Pi^2/(2+Pi)
1/2...2/Pi....A197725........ x = Pi^2/(4+Pi)
1/2...Pi/2....A197726........ x = Pi/(1+Pi)
1/2...Pi/4....A197727........ x = 2*Pi/(2+Pi)
1/3...Pi/3....A197728........ x = 3*Pi/(2+2*Pi)
1/3...Pi/6....A197729........ x = 3*Pi/(2+Pi)
2/3...Pi/6....A197730........ x = 3*Pi/(4+Pi)
1/4...Pi......A197731........ x = 2*Pi/(1+4*Pi)
1/4...Pi/2....A197732........ x = 2*Pi/(1+2*Pi)
1/4...Pi/4....A197733........ x = 2*Pi/(1+Pi)
1/5...Pi/5....10*A197691..... x = 5*Pi/(2+2*Pi)
1/6...Pi/6....A197735........ x = 3*Pi/(1+Pi)
1/8...Pi/8....A197736........ x = 4*Pi/(1+Pi)

Examples

			0.37927349649738807267221534452244643...

Links

Crossrefs

Cf. A197683.

Programs

  • Mathematica
    b = 1; c = Pi;
t = x /. FindRoot[Sin[b*x] == Cos[c*x], {x, .3, .4}]
N[Pi/(2*b + 2*c), 110]
RealDigits[%]  (* A197682 *)
Simplify[Pi/(2*b + 2*c)]
Plot[{Sin[b*x], Cos[c*x]}, {x, 0, Pi}]
  • PARI
    1/(2/Pi+2) \\ Charles R Greathouse IV, Sep 27 2022

A195057 Decimal expansion of Pi^2/11.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Oct 04 2011

Keywords

Examples

			0.8972367637353962380758628181705591941194...

Links

Crossrefs

Cf. A002388, A013661, A019678, A072691, A100044, A195059.

Programs

  • Magma
    Pi(RealField(129))^2/11; // G. C. Greubel, Jun 03 2021
  • Mathematica
    RealDigits[Pi^2/11, 10, 105][[1]] (* T. D. Noe, Oct 05 2011 *)
  • Sage
    numerical_approx(pi^2/11, digits=128) # G. C. Greubel, Jun 03 2021

Extensions

Extended by T. D. Noe, Oct 05 2011

A387440 Decimal expansion of cos(Pi/11).

Original entry on oeis.org

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

Views

Author

R. J. Mathar, Aug 29 2025

Keywords

Examples

			0.95949297361449738989...

Links

Crossrefs

Cf. A019678.

Formula

Largest of the 5 real-valued roots of 32*x^5 -16*x^4 -32*x^3 +12*x^2 +6*x-1 =0.
Equals sin(9*Pi/22).

A019697 Decimal expansion of 2*Pi/11.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

The famous rose window of the Troia cathedral in Apulia, Italy, consists of eleven columns that radiate from the center with unusual angles 2Pi/11 = 32.72... °. - Jean-François Alcover, May 25 2015

Links

Crossrefs

Equals twice A019678.

Programs

A342977 Decimal expansion of (Pi - 2) / 4.

Original entry on oeis.org

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

Views

Author

Michal Paulovic, Apr 01 2021

Keywords

Comments

The constant represents the area of a circular segment bounded by an arc of Pi/2 radians (the right angle) of a unit circle and by a chord of the length of sqrt(2). Four such segments result when a square with the side length of sqrt(2) is circumscribed by a unit circle. The area of each segment is:
A = (R^2 / 2) * (theta - sin(theta))
A = (1^2 / 2) * (Pi/2 - sin(Pi/2))
A = (1 / 2) * (Pi/2 - 1)
A = (Pi - 2) / 4 = 0.28539816...
where Pi = 3.14159265... (A000796) is the area bounded by the unit circle, and 2 is the area of the inscribed square.
Apart from the first digit this is the same as Pi/4 = 0.78539816... (A003881), the area of a circular sector bounded by the arc of Pi/2 = 1.57079632... (A019669) radians of the unit circle and by two radii of unit length, and 1/2 = 0.5 (A020761) is one-quarter of the area of the inscribed square.
The constant is close to 2/7 = 0.28571428... (2 * A020806) and Pi/11 = 0.28559933... (A019678). The equation (x - 2)/4 = x/11 has a solution x = 22/7 = 3.14285714... (A068028), which is an approximation of Pi.
The best rational approximation of the constant using small positive integers (less than 1000) is 129/452 = 0.28539823..., the next best approximation is 4771/16717 = 0.28539809...
The reciprocal of the constant is:
1/A = 4 / (Pi - 2) = 3.50387678... (A309091).
The sagitta (height) of the circular segment is:
h = R * (1 - cos(theta/2))
h = 1 * (1 - cos(Pi/4))
h = 1 - sqrt(2) / 2
h = 1 - 1 / sqrt(2) = 0.29289321... (A268682).

Examples

			0.2853981633974483...

Links

Crossrefs

Cf. A000796, A019669, A019678, A020761, A020806, A068028, A268682, A309091. Essentially the same as A003881.

Programs

  • Mathematica
    RealDigits[Pi/4 - 1/2, 10, 100][[1]] (* Amiram Eldar, Jun 08 2021 *)
  • PARI
    (Pi - 2) / 4

Formula

Equals Integral_{x=-sqrt(2)/2..sqrt(2)/2} Integral_{y=sqrt(2)/2..sqrt(1-x^2)} dy dx.
Equals Sum_{k>=1} (-1)^(k + 1)/(4*k^2 - 1). - Amiram Eldar, Jun 08 2021
Continued fraction: 1/(3 + 3/(4 + 15/(4 + 35/(4 + ... + (4*n^2 - 1)/(4 + ...). - Peter Bala, Feb 22 2024
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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