A140239 -id:A140239 - cp's OEIS frontend

A140244 Decimal expansion of arccos(-1/4).

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

Offset: 1

Rick L. Shepherd, May 14 2008

Angle in radians of the obtuse angle of the obtuse scalene triangle with sides of lengths 2, 3 and 4, the scalene triangle with least integer side lengths.
A140240 + A140242 + A140244 = arccos(7/8) + arccos(11/16) + arccos(-1/4) = Pi.
Arccos(-1/4) is the least positive x for which the function f(x)=cos(x)+cos(2x) attains its minimum value, which is -9/8. - Clark Kimberling, Oct 28 2011

			1.82347658193697527271697912863346241435077843278439110412139607489448326362...

Index entries for transcendental numbers

Cf. A140239, A140240, A140241, A140242, A140243, A140245, A140246, A140247, A140248, A140249.

RealDigits[ArcCos[-1/4],10,120][[1]] (* Harvey P. Dale, Dec 20 2016 *)

PARI
```
acos(-1/4)
```

arccos(-1/4) = Pi - arcsin(sqrt(15)/4) = Pi - arctan(sqrt(15)).

A140246 Decimal expansion of sqrt(15)/6.

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, May 14 2008

Inradius of the obtuse scalene triangle with sides of lengths 2, 3 and 4, the scalene triangle with least integer side lengths. Per the Weisstein link, the inradius is the area divided by the semiperimeter.

			0.64549722436790281419654423329706660180548695088193180443126229435224718198...

Daniel Starodubtsev, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Inradius.
Index entries for algebraic numbers, degree 2

Cf. A140239, A140240, A140241, A140242, A140243, A140244, A140245, A140247, A140248, A140249, A088543, A010472.

Equals sqrt(A331257(8)/A331258(8)) (squared inradii of triangles with integer sides).

RealDigits[Sqrt[15]/6,10,120][[1]] (* Harvey P. Dale, Mar 31 2013 *)

PARI
```
sqrt(15)/6
```

sqrt(15)/6 = A010472/6 = 2*A140239/9.

A140248 Decimal expansion of 0.3 * sqrt(15).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, May 14 2008

Exradius opposite the side of length 2 of the obtuse scalene triangle with sides of lengths 2, 3 and 4, the scalene triangle with least integer side lengths. See formulas in the Weisstein link.

Multiplied by 10, this is sqrt(135). - Alonso del Arte, Jan 06 2013

			1.16189500386222506555377961993471988324987651158747724797627212983404492758...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Exradius.
Index entries for algebraic numbers, degree 2

Cf. A140239, A140240, A140241, A140242, A140243, A140244, A140245, A140246, A140247, A140249, A088543, A010472.

RealDigits[(3/10)Sqrt[15], 10, 105][[1]] (* Alonso del Arte, Jan 06 2013 *)

PARI
```
0.3*sqrt(15)
```

0.3*sqrt(15) = 0.3*A010472 = 0.4*A140239 = 0.6*A088543 = 0.2*A140249.

A140245 Decimal expansion of 180*arccos(-1/4)/Pi.

Original entry on oeis.org

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

Offset: 3

Rick L. Shepherd, May 14 2008

Angle in degrees of the obtuse angle of the obtuse scalene triangle with sides of lengths 2, 3 and 4, the scalene triangle with least integer side lengths.

A140241 + A140243 + A140245 = 180.

			104.477512185929923878771034799127166005131597624556616476050118008851293580...

Cf. A140239, A140240, A140241, A140242, A140243, A140244, A140246, A140247, A140248, A140249.

PARI
```
180*acos(-1/4)/Pi
```

180*arccos(-1/4)/Pi = 180*A140244/Pi.

A140247 Decimal expansion of 8/sqrt(15).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, May 14 2008

Circumradius of the obtuse scalene triangle with sides of lengths 2, 3 and 4, the scalene triangle with least integer side lengths. See formulas in the Weisstein link.

			2.06559111797728900542894154655061312577755824282218177418003934192719098236...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Circumradius.
Index entries for algebraic numbers, degree 2

Cf. A140239, A140240, A140241, A140242, A140243, A140244, A140245, A140246, A140248, A140249, A088543, A010472.

Equals sqrt(A331227(10)/A331228(10)) = sqrt(A331227(11)/A331228(11)), A331254, A331255, A331256 (list of triangles with integer sides sorted by circumradius).

RealDigits[8/Sqrt[15],10,120][[1]] (* Harvey P. Dale, May 06 2012 *)

PARI
```
8/sqrt(15)
```

8/sqrt(15) = 8/A010472.

A140249 Decimal expansion of 3*sqrt(15)/2.

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, May 14 2008

Exradius opposite the side of length 4 of the obtuse scalene triangle with sides of lengths 2, 3 and 4, the scalene triangle with least integer side lengths. See formulas in the Weisstein link.

			5.80947501931112532776889809967359941624938255793738623988136064917022463790...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Exradius.
Index entries for algebraic numbers, degree 2

Cf. A140239, A140240, A140241, A140242, A140243, A140244, A140245, A140246, A140247, A140248, A088543, A010472.

First[RealDigits[3 Sqrt[15]/2,10,100]] (* Paolo Xausa, Oct 30 2023 *)

PARI
```
3*sqrt(15)/2
```

3*sqrt(15)/2 = 3*A010472/2 = 2*A140239 = 3*A088543 = 5*A140248.

A140240 Decimal expansion of arccos(7/8).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, May 14 2008

Angle in radians of the least angle of the obtuse scalene triangle with sides of lengths 2, 3 and 4, the scalene triangle with least integer side lengths.

A140240 + A140242 + A140244 = arccos(7/8) + arccos(11/16) + arccos(-1/4) = Pi.

			0.50536051028415730697131487398742194450438746619367638726784755748115012096...

Index entries for transcendental numbers

Cf. A140239, A140241, A140242, A140243, A140244, A140245, A140246, A140247, A140248, A140249.

RealDigits[ArcCos[7/8],10,120][[1]] (* Harvey P. Dale, Jan 13 2019 *)

PARI
```
acos(7/8)
```

arccos(7/8) = arcsin(sqrt(15)/8) = arctan(sqrt(15)/7).

A140241 Decimal expansion of 180*arccos(7/8)/Pi.

Original entry on oeis.org

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

Offset: 2

Rick L. Shepherd, May 14 2008

Angle in degrees of the least angle of the obtuse scalene triangle with sides of lengths 2, 3 and 4, the scalene triangle with least integer side lengths.

A140241 + A140243 + A140245 = 180.

			28.9550243718598477575420695982543320102631952491132329521002360177025871614...

Cf. A140239, A140240, A140242, A140243, A140244, A140245, A140246, A140247, A140248, A140249.

RealDigits[(180*ArcCos[7/8])/Pi,10,120][[1]] (* Harvey P. Dale, Jul 19 2014 *)

PARI
```
180*acos(7/8)/Pi
```

180*arccos(7/8)/Pi = 180*A140240/Pi.

A140242 Decimal expansion of arccos(11/16).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, May 14 2008

Angle in radians of the larger acute angle of the obtuse scalene triangle with sides of lengths 2, 3 and 4, the scalene triangle with least integer side lengths.

A140240 + A140242 + A140244 = arccos(7/8) + arccos(11/16) + arccos(-1/4) = Pi.

			0.81275556136866065877434938065861852534200350039703832958570095993218302170...

Index entries for transcendental numbers

Cf. A140239, A140240, A140241, A140243, A140244, A140245, A140246, A140247, A140248, A140249.

RealDigits[ArcCos[11/16],10,120][[1]] (* Harvey P. Dale, Aug 15 2024 *)

PARI
```
acos(11/16)
```

arccos(11/16) = arcsin(3*sqrt(15)/16) = arctan(3*sqrt(15)/11).

A140243 Decimal expansion of 180*arccos(11/16)/Pi.

Original entry on oeis.org

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

Offset: 2

Rick L. Shepherd, May 14 2008

Angle in degrees of the larger acute angle of the obtuse scalene triangle with sides of lengths 2, 3 and 4, the scalene triangle with least integer side lengths.

A140241 + A140243 + A140245 = 180.

			46.5674634422102283636868956026185019846052071263301505718496459734461192578...

Cf. A140239, A140240, A140241, A140242, A140244, A140245, A140246, A140247, A140248, A140249.

RealDigits[180 ArcCos[11/16]/Pi,10,120][[1]] (* Harvey P. Dale, Jul 07 2022 *)

PARI
```
180*acos(11/16)/Pi
```

180*arccos(11/16)/Pi = 180*A140242/Pi.

cp's OEIS Frontend

A140244 Decimal expansion of arccos(-1/4).

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A140246 Decimal expansion of sqrt(15)/6.

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A140248 Decimal expansion of 0.3 * sqrt(15).

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A140245 Decimal expansion of 180*arccos(-1/4)/Pi.

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A140247 Decimal expansion of 8/sqrt(15).

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A140249 Decimal expansion of 3*sqrt(15)/2.

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A140240 Decimal expansion of arccos(7/8).

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