A140244 -id:A140244 - cp's OEIS frontend

A140239 Decimal expansion of 3*sqrt(15)/4.

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

Offset: 1

Rick L. Shepherd, May 14 2008

Area of the obtuse scalene triangle with sides of lengths 2, 3 and 4, the scalene triangle with least integer side lengths.

This is the area of the ninth-smallest triangle with integer side lengths, or the eighth-smallest triangle if two smaller triangles with the same area are counted only once (see A331251). - Hugo Pfoertner, Feb 12 2020

			2.90473750965556266388444904983679970812469127896869311994068032458511231895...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 2

Cf. A140240, A140241, A140242, A140243, A140244, A140245, A140246, A140247, A140248, A140249, A088543, A010472, A331250, A331251.

RealDigits[(3Sqrt[15])/4,10,120][[1]] (* Harvey P. Dale, Apr 03 2013 *)

3*sqrt(15)/4 = 3*A010472/4.

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.

A129556 Numbers k such that the k-th centered pentagonal number A005891(k) = (5k^2 + 5k + 2)/2 is a square.

Original entry on oeis.org

0, 2, 21, 95, 816, 3626, 31005, 137711, 1177392, 5229410, 44709909, 198579887, 1697799168, 7540806314, 64471658493, 286352060063, 2448225223584, 10873837476098, 92968086837717, 412919472031679, 3530339074609680, 15680066099727722, 134059916748330141

Offset: 1

Alexander Adamchuk, Apr 20 2007

Corresponding numbers m > 0 such that m^2 is a centered pentagonal number are listed in A129557 = {1, 4, 34, 151, 1291, 5734, 49024, ...}.
From Andrea Pinos, Nov 02 2022: (Start)
By definition: 5*T(a(n)) = A129557(n)^2 - 1 where triangular number T(j) = j*(j+1)/2. This implies:
Every odd prime factor of a(n) and d(n)=a(n)+1 is present in b(n)=A129557(n)+1 or in c(n)=A129557(n)-1. (End)
From the law of cosines the non-Pythagorean triple {a(n), a(n)+1=A254332(n), A129557(n+1)} forms a near-isosceles triangle whose angle between the consecutive integer sides is equal to the central angle of the regular pentachoron polytope (4-simplex) (see A140244 and A140245). This implies that the terms {a(n)} are also those numbers k such that 1 + 5*A000217(k) is a square. - Federico Provvedi, Apr 04 2023

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Centered Pentagonal Number
Index entries for linear recurrences with constant coefficients, signature (1,38,-38,-1,1).

Cf. A005891 (centered pentagonal numbers), A129557 (numbers k>0 such that k^2 is a centered pentagonal number), A221874.

Cf. numbers m such that k*A000217(m)+1 is a square: A006451 for k=1; m=0 for k=2; A233450 for k=3; A001652 for k=4; this sequence for k=5; A001921 for k=6. - Bruno Berselli, Dec 16 2013

A005891 := proc(n) (5*n^2+5*n+2)/2 ; end: n := 0 : while true do if issqr(A005891(n)) then print(n) ; fi ; n := n+1 ; od : # R. J. Mathar, Jun 06 2007

Do[ f=(5n^2+5n+2)/2; If[ IntegerQ[ Sqrt[f] ], Print[n] ], {n,1,40000} ]
LinearRecurrence[{1,38,-38,-1,1},{0,2,21,95,816},30] (* Harvey P. Dale, Nov 09 2017 *)
Table[(((x^(n+2))+(((-1)^n*(x^(2*n+1)+1)-x)/(x^n)))/(x^2+1)-1)/2/.x->3+Sqrt[10],{n,0,50}]//Round (* Federico Provvedi, Apr 04 2023 *)

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; 1,-1,-38,38,1]^(n-1)*[0;2;21;95;816])[1,1] \\ Charles R Greathouse IV, Feb 11 2019

For n >= 5, a(n) = 38*a(n-2) - a(n-4) + 18. - Max Alekseyev, May 08 2009
G.f.: x^2*(x^3+2*x^2-19*x-2) / ((x-1)*(x^2-6*x-1)*(x^2+6*x-1)). - Colin Barker, Feb 21 2013
a(n) = (A221874(n) - 1) / 2. - Bruno Berselli, Feb 21 2013
From Andrea Pinos, Oct 24 2022: (Start)
The ratios of successive terms converge to two different limits:
lower: D = lim_{n->oo} a(2n)/a(2n-1) = (7+2*sqrt(10))/3;
upper: E = lim_{n->oo} a(2n+1)/a(2n) = (13+4*sqrt(10))/3.
So lim_{n->oo} a(n+2)/a(n) = D*E = 19 + 6*sqrt(10).  (End)
a(n) = (x^(2*(n+1)) + (-1)^n*(x^(2*n+1)+1) - x) / (2*x^n*(x^2 + 1)) - (1/2), with x=3+sqrt(10). - Federico Provvedi, Apr 04 2023

More terms from R. J. Mathar, Jun 06 2007
Further terms from Max Alekseyev, May 08 2009
a(22)-a(23) from Colin Barker, Feb 21 2013

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).

cp's OEIS Frontend

A140239 Decimal expansion of 3*sqrt(15)/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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A129556 Numbers k such that the k-th centered pentagonal number A005891(k) = (5k^2 + 5k + 2)/2 is a square.

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