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.

A179378 Decimal expansion of the ratio of the area of the triangle corresponding to a circular segment with area r^2 of a circle with radius r to r^2 itself.

Original entry on oeis.org

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

Views

Author

Rick L. Shepherd, Jul 11 2010

Keywords

Comments

In other words, the triangle area is A179378*(r^2). The triangle height is A179377*r. The segment height ("cap height" in MathWorld link) is A179376*r. The chord length is A179375*r. The arc length of the circular segment/sector is r*A179373. The area of the circular segment, r^2, is 1/Pi (A049541) times the area of the circle. The area of the sector is (r^2)*(A179373/2) = (r^2)*(1 + A179378). See references and cross-references for other relationships.
From Clark Kimberling, Oct 10 2011: (Start)
Decimal expansion of xo, where P=(xo,yo) is the point nearest O=(0,0) in which a line y=mx meets the curve y=cos(x+1) orthogonally; specifically:
xo=0.277097976418521518914833086895...
yo=0.289494183027862650094360757305...
m=1.0447358251025919644670467125044...
|OP|=0.4007370341535820008719293563... See the Mathematica program for a graph. (End)

Examples

			.2770979764185215189148330868959389680578745857055262190702831821510113134466...

References

  • S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 7.

Links

Crossrefs

Cf. A179373 (central angle, radians), A179374 (central angle, degrees), A179375 (for chord length), A179376 (for "cap height", height of segment), A179377 (for triangle height), A049541; A197009 (radius of orthogonal circle).

Programs

  • Mathematica
    c = 1;
xo = x /. FindRoot[x == Sin[x + c] Cos[x + c], {x, .8, 1.2}, WorkingPrecision -> 100]
RealDigits[xo] (* A179378 *)
m = 1/Sin[xo + c]
RealDigits[m]  (* A197009 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{Cos[x + c], yo - (1/m) (x - xo)}, {x, -Pi/4, Pi/2}],
ContourPlot[{y == m*x}, {x, 0, Pi}, {y, 0, 1}], PlotRange -> All,
AspectRatio -> Automatic, AxesOrigin -> Automatic]
  • PARI
    sin(solve(x=0, Pi, x-sin(x)-2))/2

Formula

Equals sin(A179373)/2 = sin(A179373/2)*cos(A179373/2) = A179375*A179377/2.

A179373 Decimal expansion of the central angle in radians corresponding to a circular segment with area r^2 of a circle with radius r.

Original entry on oeis.org

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

Views

Author

Rick L. Shepherd, Jul 11 2010

Keywords

Comments

The arc length of the circular segment/sector is r*A179373. The area of the circular segment, r^2, is 1/Pi (A049541) times the area of the circle. The area of the sector is (r^2)*(A179373/2) = (r^2)*(1 + A179378). See references and cross-references for other relationships.

Examples

			2.5541959528370430378296661737918779361157491714110524381405663643020...

References

  • S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 7.

Links

Crossrefs

Cf. A179374 (same, in degrees), A179375 (for chord length), A179376 (for "cap height", height of segment), A179377 (for triangle height), A179378 (for triangle area), A049541.

Programs

  • Mathematica
    RealDigits[ x /. FindRoot[x - Sin[x] - 2, {x, 2}, WorkingPrecision -> 105]][[1]] (* Jean-François Alcover, Oct 30 2012 *)
  • PARI
    solve(x=0, Pi, x-sin(x)-2)

Formula

Decimal expansion of the solution of sin(x) = x - 2.

A179374 Decimal expansion of the central angle in degrees corresponding to a circular segment with area r^2 of a circle with radius r.

Original entry on oeis.org

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

Views

Author

Rick L. Shepherd, Jul 11 2010

Keywords

Comments

The arc length of the circular segment/sector is r*A179373. The area of the circular segment, r^2, is 1/Pi (A049541) times the area of the circle. The area of the sector is (r^2)*(A179373/2) = (r^2)*(1 + A179378). See references and cross-references for other relationships.

Examples

			146.3446481469584283647321150080224451316690962652634500095885765914885737881...

References

  • S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 7.

Links

Crossrefs

Cf. A179373 (same, in radians), A179375 (for chord length), A179376 (for "cap height", height of segment), A179377 (for triangle height), A179378 (for triangle area), A049541.

Programs

  • Mathematica
    RealDigits[(180/Pi)*(x /.FindRoot[x-Sin[x]-2, {x, 2}, WorkingPrecision -> 200]), 10, 100][[1]] (* G. C. Greubel, Nov 16 2018 *)
  • PARI
    (solve(x=0, Pi, x-sin(x)-2))*180/Pi

Formula

Equals A179373*180/Pi = A179373*A072097.

A179375 Decimal expansion of the ratio of the chord length of a circular segment with area r^2 of a circle with radius r to r itself.

Original entry on oeis.org

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

Views

Author

Rick L. Shepherd, Jul 11 2010

Keywords

Comments

In other words, the chord length is A179375*r. The arc length of the circular segment/sector is r*A179373. The area of the circular segment, r^2, is 1/Pi (A049541) times the area of the circle. The area of the sector is (r^2)*(A179373/2) = (r^2)*(1 + A179378). See references and cross-references for other relationships.

Examples

			1.914359546159529939985785242460527899501301180791115677192453168859627644264...

References

  • S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 7.

Links

Crossrefs

Cf. A179373 (central angle, radians), A179374 (central angle, degrees), A179376 (for "cap height", height of segment), A179377 (for triangle height), A179378 (for triangle area), A049541.

Programs

  • Mathematica
    RealDigits[ 2*Sin[x/2] /. FindRoot[x - Sin[x] - 2, {x, 2}, WorkingPrecision -> 106]][[1]] (* Jean-François Alcover, Oct 30 2012 *)
  • PARI
    2*sin(solve(x=0, Pi, x-sin(x)-2)/2)

Formula

Equals 2*sin(A179373/2).

A179376 Decimal expansion of the ratio of the height of a circular segment with area r^2 of a circle with radius r to r itself.

Original entry on oeis.org

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

Views

Author

Rick L. Shepherd, Jul 11 2010

Keywords

Comments

In other words, the segment height ("cap height" in MathWorld link) is A179376*r. The chord length is A179375*r. The arc length of the circular segment/sector is r*A179373. The area of the circular segment, r^2, is 1/Pi (A049541) times the area of the circle. The area of the sector is (r^2)*(A179373/2) = (r^2)*(1 + A179378). See references and cross-references for other relationships.

Examples

			.71050581697213734990563924269484526760618954800103872979253477385591...

References

  • S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 7.

Links

Crossrefs

Cf. A179373 (central angle, radians), A179374 (central angle, degrees), A179375 (for chord length), A179377 (for triangle height), A179378 (for triangle area), A133742, A049541.

Programs

  • Mathematica
    RealDigits[1-x /. FindRoot[x == Cos[1+x*Sqrt[1-x^2]], {x, 0}, WorkingPrecision -> 120]][[1]] (* Jean-François Alcover, Oct 06 2011 *)
  • PARI
    1 - cos(solve(x=0, Pi, x-sin(x)-2)/2)

Formula

Equals 1 - cos(A179373/2) = 1 - A179377.
Showing 1-5 of 5 results.

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