A049541 -id:A049541 - cp's OEIS frontend

A032615 a(n) = floor(n/Pi).

0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24

Offset: 0

Patrick De Geest, May 15 1998

a(n) = A062300(n-1) = floor(1/sin(Pi/n)) for almost all indices, the next exception after n = 6 being n = 80143857, cf. link to Kevin Ryde's post to the SeqFan list. - M. F. Hasler, Oct 20 2016

R. Israel, in reply to K. Ryde, Re: nearly equal floor(n/Pi) A032615 and A062300, SeqFan list, Oct 19 2016.
Index entries for sequences related to Beatty sequences

Cf. A062300. Beatty seq. of A049541.

A032615:=n->floor(n/Pi); seq(A032615(n), n=0..100); # Wesley Ivan Hurt, Dec 10 2013

Table[Floor[n/Pi], {n, 0, 100}] (* Wesley Ivan Hurt, Dec 10 2013 *)

a(n)=n\Pi \\ Charles R Greathouse IV, Dec 10 2013

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

Rick L. Shepherd, Jul 11 2010

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)

			.2770979764185215189148330868959389680578745857055262190702831821510113134466...

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

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Circular Segment.

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).

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
```

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

A073238 Decimal expansion of Pi^(1/Pi).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jul 25 2002

Pi^(1/Pi) = 1/((1/Pi)^(1/Pi)) (reciprocal of A073240).

			1.43961949584759068833649080497...

Cf. A000796 (Pi), A049541 (1/Pi), A073239 ((1/Pi)^Pi), A073240 ((1/Pi)^(1/Pi)), A073233 (Pi^Pi).

RealDigits[N[Pi^(1/Pi),200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2012 *)

PARI
```
Pi^(1/Pi)
```

A073240 Decimal expansion of (1/Pi)^(1/Pi).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jul 27 2002

(1/Pi)^(1/Pi) = Pi^(-1/Pi) = 1/(Pi^(1/Pi)) (reciprocal of A073238).

			0.69462799224682615312438376141...

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A000796 (Pi), A049541 (1/Pi), A073241 ((1/Pi)^(1/Pi)^(1/Pi)), A073243 (limit of (1/Pi)^(1/Pi)^...^(1/Pi)), A073238 (Pi^(1/Pi)), A073239 ((1/Pi)^Pi), A073233 (Pi^Pi).

First[RealDigits[(1/Pi)^(1/Pi),10,100]] (* Paolo Xausa, Nov 07 2023 *)

PARI
```
(1/Pi)^(1/Pi)
```

A165952 Decimal expansion of 2sqrt(3)/(3Pi).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Oct 02 2009

The ratio of the volume of a cube to the volume of the circumscribed sphere (which has circumradius a*sqrt(3)/2 = a*A010527, where a is the cube's edge length; see MathWorld link). For similar ratios for other Platonic solids, see A165922, A049541, A165953, and A165954. A063723 shows the order of these by size.

			0.3675525969478613663408843322086462942649243202444271018662440135273535356...

Eric Weisstein's World of Mathematics, Cube.
Index entries for transcendental numbers

Cf. A000796, A002194, A165922, A049541, A165953, A165954, A063723, A020760, A020832.

RealDigits[(2*Sqrt[3])/(3Pi),10,120][[1]] (* Harvey P. Dale, Oct 08 2012 *)

PARI
```
2*sqrt(3)/(3*Pi)
```

2*sqrt(3)/(3*Pi) = 2*A002194/(3*A000796) = 3*A165922 = (2*sqrt(3)/3)*A049541 = 10*A020832*A049541 = 2*A020760*A049541.

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

Rick L. Shepherd, Jul 11 2010

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.

			2.5541959528370430378296661737918779361157491714110524381405663643020...

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

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Circular Segment.
Index entries for transcendental numbers.

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.

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)
```

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

Rick L. Shepherd, Jul 11 2010

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.

			146.3446481469584283647321150080224451316690962652634500095885765914885737881...

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

G. C. Greubel, Table of n, a(n) for n = 3..10002
Eric Weisstein's World of Mathematics, Circular Segment.

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.

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
```

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

Rick L. Shepherd, Jul 11 2010

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.

			1.914359546159529939985785242460527899501301180791115677192453168859627644264...

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

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Circular Segment.
Index entries for transcendental numbers.

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.

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)
```

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

Rick L. Shepherd, Jul 11 2010

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.

			.71050581697213734990563924269484526760618954800103872979253477385591...

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

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Circular Segment.

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

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)
```

Equals 1 - cos(A179373/2) = 1 - A179377.

A179377 Decimal expansion of the ratio of the height of the triangle corresponding to a circular segment with area r^2 of a circle with radius r to r itself.

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jul 11 2010

In other words, 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.

			.2894941830278626500943607573051547323938104519989612702074652261440891212633...

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

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Circular Segment.

Cf. A179373 (central angle, radians), A179374 (central angle, degrees), A179375 (for chord length), A179376 (for "cap height", height of segment), A179378 (for triangle area), A049541.

RealDigits[ Cos[x/2] /. FindRoot[x - Sin[x] - 2, {x, 1}, WorkingPrecision -> 106]][[1]] (* Jean-François Alcover, Oct 30 2012 *)

PARI
```
cos(solve(x=0, Pi, x-sin(x)-2)/2)
```

Equals cos(A179373/2) = 1 - A179376.

cp's OEIS Frontend

A032615 a(n) = floor(n/Pi).

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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.

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A073238 Decimal expansion of Pi^(1/Pi).

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A073240 Decimal expansion of (1/Pi)^(1/Pi).

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Mathematica

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A165952 Decimal expansion of 2*sqrt(3)/(3*Pi).

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

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Mathematica

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

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A165952 Decimal expansion of 2sqrt(3)/(3Pi).