A093064 -id:A093064 - cp's OEIS frontend

A093063 Decimal expansion of (2 + 4sqrt(2) + (4 + sqrt(2))arcsinh(1))/30.

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

Offset: 0

Eric W. Weisstein, Mar 16 2004

Average length of a line picked at random inside an isosceles right triangle with legs of unit length.

Close but not equal to sqrt(2)-1 = 0.414213562.

			0.41429330259503377743084107857530325185642398130824894805...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Triangle Line Picking
Index entries for transcendental numbers

Cf. A093064.

SetDefaultRealField(RealField(100)); (2+4*Sqrt(2)+(4+Sqrt(2) )*Argsinh(1))/30; // G. C. Greubel, Sep 27 2018

RealDigits[(2+4Sqrt[2]+(4+Sqrt[2])ArcSinh[1])/30,10,120][[1]] (* Harvey P. Dale, Jan 20 2012 *)

default(realprecision, 100); (2+4*sqrt(2)+(4+sqrt(2))*asinh(1))/30 \\ G. C. Greubel, Sep 27 2018

A330646 Decimal expansion of the mean length of a line segment drawn by picking a random point on a unit length equilateral triangle and extending it in a random direction until it meets an edge.

Original entry on oeis.org

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

Offset: 0

Muthu Veerappan Ramalingam, Dec 22 2019

			0.302848349804097933375691303492564570884596990578285814558...

Muthu Veerappan Ramalingam, An Expected Value Problem III

Cf. A093064.

RealDigits[Sqrt[3] Log[3]/2/Pi, 10, 110][[1]]

Equals sqrt(3) * log(3) / (2 * Pi).

Equals (h_a/(3*Pi))*cosech^{-1}((1/2)*((b+c)/a-a/(b+c)))+(h_b/(3*Pi))*cosech^{-1}((1/2)*((c+a)/b-b/(c+a)))+(h_c/(3*Pi))*cosech^{-1}((1/2)*((a+b)/c-c/(a+b))) for an arbitrary triangle with sides a, b and c with corresponding altitudes h_a, h_b and h_c.

A180308 Decimal expansion of the mean length of a line segment picked at random in a 30-60-90 (right) triangle.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Aug 25 2010

			0.28857171834839485625...

Eric Weisstein's World of Mathematics, 30-60-90 Triangle.
Eric Weisstein's World of Mathematics, Triangle Line Picking.

Cf. A093063, A093064, A180307.

RealDigits[(204 + 36*Sqrt[3] + 81*Log[3] + 2*(9 + 8*Sqrt[3])*Log[2 + Sqrt[3]])/1440, 10, 120][[1]] (* Amiram Eldar, May 25 2023 *)

Equals (204 + 36*sqrt(3) + 81*log(3) + 2*(9 + 8*sqrt(3))*log(2 + sqrt(3)))/1440.

A382821 Decimal expansion of (3/2) * (log(3) - 1).

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Apr 05 2025

			0.147918433002164537092867855...

I. S. Gradsteyn and I. M. Ryzhik, Table of integrals, series and products (6th ed.), 2000, (eq. 0.236.2).

Cf. A016627, A016631, A145425.

Essentially the same as A093064.

RealDigits[(3/2) * (Log[3] - 1), 10, 120][[1]] (* Amiram Eldar, Aug 08 2025 *)

Equals Sum_{k>=1} 1/(k*(9*k^2-1)).

Equals Sum_{k>=1} zeta(2*k+1)/9^k. - Amiram Eldar, Aug 08 2025

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A093063 Decimal expansion of (2 + 4*sqrt(2) + (4 + sqrt(2))*arcsinh(1))/30.

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A330646 Decimal expansion of the mean length of a line segment drawn by picking a random point on a unit length equilateral triangle and extending it in a random direction until it meets an edge.

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A180308 Decimal expansion of the mean length of a line segment picked at random in a 30-60-90 (right) triangle.

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A382821 Decimal expansion of (3/2) * (log(3) - 1).

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A093063 Decimal expansion of (2 + 4sqrt(2) + (4 + sqrt(2))arcsinh(1))/30.