A093063 -id:A093063

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A093064

Decimal expansion of (4 + 3*log(3))/20.

+10
4

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

Average length of a line picked at random inside an equilateral triangle with unit side lengths.

LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Uwe Bäsel, The moments of the distance between two random points in a regular polygon, arXiv:2101.03815 [math.PR], 2021.

Eric Weisstein's World of Mathematics, Triangle Line Picking

Index entries for transcendental numbers

EXAMPLE

0.364791843...

MATHEMATICA

RealDigits[(4 + 3 Log[3])/20, 10, 105][[1]] (* Michael De Vlieger, Apr 22 2021 *)

PROG

(PARI) (log(27)+4)/20 \\ Charles R Greathouse IV, May 15 2019

CROSSREFS

Cf. A093063.

KEYWORD

nonn,cons

AUTHOR

Eric W. Weisstein, Mar 16 2004

STATUS

approved

A180307

Decimal expansion of the mean length of a line segment picked at random in a 3, 4, 5 (right) triangle.

+10
2

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, 3, 4, 5 Triangle

Eric Weisstein's World of Mathematics, Triangle Line Picking

FORMULA

Equals (20460 + 9728*log(2) + 5103*log(3))/22500.

Equals (a^3 + b^3 + 2*c^3) / (15*c^2) + (a^2 / (15*b)) * (1 + (b/c)^3) * cosech^{-1}(a/b) + (b^2 / (15*a)) * (1 + (a/c)^3) * cosech^{-1}(b/a) for an arbitrary right angled triangle with sides a, b and (hypotenuse) c. - Muthu Veerappan Ramalingam, Dec 18 2019

EXAMPLE

1.4581846347360227433...

MAPLE

evalf( (20460+9728*log(2)+5103*log(3))/22500, 111); # G. C. Greubel, Dec 20 2019

MATHEMATICA

F[a_, b_, c_]:= (a^3 +b^3 +2*c^3)/(15*c^2) +(a^2/(15*b))*(1 + (b/c)^3)* ArcCsch[a/b] +(b^2/(15*a))*(1 +(a/c)^3)*ArcCsch[b/a]; RealDigits[F[3, 4, 5], 10, 110][[1]] (* G. C. Greubel, Dec 20 2019 *)

PROG

(PARI) arcsch(z)=log(1/z+sqrt(1/z^2+1));

seglen(a, b)={my(c=sqrt(a^2+b^2)); (a^3+b^3+2*c^3)/(15*c^2)+(a^2/(15*b))*(1+(b/c)^3)*arcsch(a/b)+(b^2/(15*a))*(1+(a/c)^3)*arcsch(b/a)};

seglen(3, 4) \\ Hugo Pfoertner, Dec 18 2019

(Magma) SetDefaultRealField(RealField(111)); (20460 +9728*Log(2) +5103*Log(3) )/22500; // G. C. Greubel, Dec 20 2019

(Sage)

def F(a, b, c): return (a^3 + b^3 + 2*c^3)/(15*c^2) + (a^2/(15*b))*(1 + (b/c)^3)*arccsch(a/b) + (b^2/(15*a))*(1 + (a/c)^3)*arccsch(b/a)

numerical_approx(F(3, 4, 5), digits=110) # G. C. Greubel, Dec 20 2019

CROSSREFS

Cf. A093063.

KEYWORD

nonn,cons

AUTHOR

Eric W. Weisstein, Aug 25 2010

STATUS

approved

A180308

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

+10
0

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Table of n, a(n) for n=0..104.

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

Eric Weisstein's World of Mathematics, Triangle Line Picking.

FORMULA

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

EXAMPLE

0.28857171834839485625...

MATHEMATICA

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 *)

CROSSREFS

Cf. A093063, A093064, A180307.

KEYWORD

nonn,cons

AUTHOR

Eric W. Weisstein, Aug 25 2010

STATUS

approved

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