A093070 -id:A093070 - cp's OEIS frontend

A093527 Denominators of even raw moments in the distribution of line lengths for lines picked at random in the unit disk.

1, 1, 3, 2, 5, 1, 7, 4, 9, 5, 11, 3, 13, 7, 1, 8, 17, 3, 19, 1, 7, 11, 23, 2, 25, 13, 27, 1, 29, 15, 31, 16, 11, 17, 5, 9, 37, 19, 39, 2, 41, 1, 43, 11, 1, 23, 47, 4, 49, 25, 17, 13, 53, 9, 55, 7, 19, 29, 59, 5, 61, 31, 21, 32, 13, 1, 67, 17, 23, 7, 71, 2, 73, 37, 5, 19, 1, 13, 79

Offset: 0

Eric W. Weisstein, Mar 30 2004

			1, 128/(45*Pi), 1, 2048/(525*Pi), 5/3, 16384/(2205*Pi), ...

Stefano Spezia, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Disk Line Picking

Cf. A093070, A093526, A093528, A093529.
Second column of A098505.
Cf. A000108.

A093527 := n -> n / igcd(n,binomial(2*n,n)): # Peter Luschny, Oct 05 2011

A093527[n_]:=Denominator[Binomial[2n,n]/n]; Array[A093527,200] (* Enrique Pérez Herrero, Oct 05 2011 *)

a(k) = Denominator[(2*Gamma[3 + n])/((2 + n)*Gamma[2 + n/2]*Gamma[3 + n/2])] for n = 2k.
From Paul Barry, Sep 11 2004: (Start)
a(n) = numerator((n+1)(n+2)/binomial(2(n+1), n+1));
a(n) = numerator(2*binomial(n+2, 2)/binomial(2(n+1), n+1)). (End)
a(n) = numerator((n+1)/C(n+1)). - Paul Barry, Nov 17 2004
a(n) = denominator(binomial(2n, n)/n). - Enrique Pérez Herrero, Oct 05 2011
a(n) = n/gcd(n,binomial(2n,n)). - Peter Luschny, Oct 05 2011
a(n) = denominator((n + 2)*binomial(2*n+3, n+1)/((n + 1)*(2*n + 3))). - Stefano Spezia, Aug 06 2022

A093526 Numerators of even raw moments in the distribution of line lengths for lines picked at random in the unit disk.

Original entry on oeis.org

1, 1, 5, 7, 42, 22, 429, 715, 4862, 8398, 58786, 52003, 742900, 1337220, 646323, 17678835, 129644790, 79606450, 1767263190, 328206021, 8155422340, 45741281820, 343059613650, 107492012277, 4861946401452, 9183676536076

Offset: 0

Eric W. Weisstein, Mar 30 2004

			1, 128/(45*Pi), 1, 2048/(525*Pi), 5/3, 16384/(2205*Pi), ...

Eric Weisstein's World of Mathematics, Disk Line Picking

Cf. A093070, A093527, A093528, A093529.

Cf. A098512, A000108.

A195686 := n -> binomial(2*n,n) / igcd(n,binomial(2*n,n));
A093526 := n -> A195686(n+1)/(n+2); # Peter Luschny, Oct 06 2011

a[n_] := Binomial[2(n+1), n+1]/((n+2) GCD[n+1, Binomial[2(n+1), n+1]]);
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jul 30 2018, after Peter Luschny *)

a(n) = denominator((n+1)*(n+2)/binomial(2*n+2, n+1)); \\ Michel Marcus, Jul 30 2018

a(k) = Numerator[(2*Gamma[3 + n])/((2 + n)*Gamma[2 + n/2]*Gamma[3 + n/2])] for n = 2k.
a(n) = denominator((n+1)/C(n+1)). - Paul Barry, Nov 17 2004
a(n) = A195686(n+1) / (n+2). - Peter Luschny, Oct 06 2011

A093528 Numerators of odd raw moments in the distribution of line lengths for lines picked at random in the unit disk.

Original entry on oeis.org

128, 2048, 16384, 524288, 4194304, 67108864, 536870912, 34359738368, 274877906944, 4398046511104, 35184372088832, 1125899906842624, 9007199254740992, 144115188075855872, 1152921504606846976

Offset: 1

Eric W. Weisstein, Mar 30 2004

			1, 128/(45*Pi), 1, 2048/(525*Pi), 5/3, 16384/(2205*Pi), ...

Eric Weisstein's World of Mathematics, Disk Line Picking

Cf. A093070, A093526, A093527, A093529.

a(k) = Numerator[(2*Gamma[3 + n])/((2 + n)*Gamma[2 + n/2]*Gamma[3 + n/2])] for n = 2k-1.

A093529 Pi*denominators of odd raw moments in the distribution of line lengths for lines picked at random in the unit disk.

Original entry on oeis.org

45, 525, 2205, 31185, 99099, 585585, 1640925, 35334585, 92147055, 468495027, 1166167275, 11408158125, 27484885575, 130734984825, 307452619485, 11455089532425, 26442675480375, 121132637200575, 275520749478975

Offset: 1

Eric W. Weisstein, Mar 30 2004

			1, 128/(45*Pi), 1, 2048/(525*Pi), 5/3, 16384/(2205*Pi), ...

Eric Weisstein's World of Mathematics, Disk Line Picking

Cf. A093070, A093526, A093527, A093528.

a(k) = Pi*Denominator[(2*Gamma[3 + n])/((2 + n)*Gamma[2 + n/2]*Gamma[3 + n/2])] for n = 2k-1.

A130203 Decimal expansion of the mean distance between two points picked at random in a half-disk.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, May 16 2007

			0.70605340938985211981936956070713315272380666325773...

Eric Weisstein's World of Mathematics, Circular Sector Line Picking.
Index entries for transcendental numbers.

Cf. A093070, A130204.

RealDigits[64*(12*Pi-23)/(135*Pi^2), 10, 100][[1]] (* Amiram Eldar, Apr 14 2022 *)

64*(12*Pi-23)/135/Pi^2 \\ Charles R Greathouse IV, Nov 26 2024

Equals (64*(-23 + 12*Pi))/(135*Pi^2).

A130204 Decimal expansion of the mean distance between two points picked at random in a quarter-disk.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, May 16 2007

			0.47387726224232582979...

Eric Weisstein's World of Mathematics, Circular Sector Line Picking.

Cf. A093070, A130203.

RealDigits[32*(3 - 94*Sqrt[2] + 48*Pi + 6*Log[2*(Sqrt[2]-1)])/(135*Pi^2), 10, 100][[1]] (* Amiram Eldar, Jun 25 2021 *)

(32*(3 - 94*sqrt(2) + 48*Pi + 6*log(-2 + 2*sqrt(2))))/(135*Pi^2) \\ Michel Marcus, Jun 25 2021

Equals (32*(3 - 94*sqrt(2) + 48*Pi + 6*log(-2 + 2*sqrt(2))))/(135*Pi^2).

A307029 Decimal expansion of the average distance of a point from a line that passes through another 2 points, when the 3 points are randomly selected within a unit-radius disk.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Mar 20 2019

			0.526998816760654618202070198485135757399687036263519...

Roger Pinkham, Problem 10585, The American Mathematical Monthly, Vol. 104, No. 5 (1997), pp. 456, solution, by Richard Holzsager, Vol. 106, No. 2 (1999), pp. 171-172.

Cf. A093070, A130203, A130204, A245684.

RealDigits[8192/(1575*Pi^2), 10, 100][[1]]

Equals 8192/(1575*Pi^2).

cp's OEIS Frontend

A093527 Denominators of even raw moments in the distribution of line lengths for lines picked at random in the unit disk.

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A093526 Numerators of even raw moments in the distribution of line lengths for lines picked at random in the unit disk.

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A093528 Numerators of odd raw moments in the distribution of line lengths for lines picked at random in the unit disk.

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A093529 Pi*denominators of odd raw moments in the distribution of line lengths for lines picked at random in the unit disk.

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A130203 Decimal expansion of the mean distance between two points picked at random in a half-disk.

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A130204 Decimal expansion of the mean distance between two points picked at random in a quarter-disk.

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A307029 Decimal expansion of the average distance of a point from a line that passes through another 2 points, when the 3 points are randomly selected within a unit-radius disk.

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