A091648 -id:A091648 - cp's OEIS frontend

A354634 Decimal expansion of the negated digamma function at 5/8.

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

Offset: 1

R. J. Mathar, Jun 01 2022

			psi(5/8) = -1.45270876457656...

Index entries for sequences related to the digamma function

Cf. A001620, A091648.

RealDigits[PolyGamma[5/8], 10, 100][[1]] (* Amiram Eldar, Jun 03 2022 *)

Equals sqrt(2)*arcsinh(1) - 4*log(2) + (sqrt(2)-1)*Pi/2 - gamma, where gamma is Euler's constant (A001620). - Amiram Eldar, Jun 03 2022

A354635 Decimal expansion of the negated digamma function at 7/8.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jun 01 2022

			psi(7/8) = -0.80401707154769...

Index entries for sequences related to the digamma function

Cf. A001620, A091648.

RealDigits[PolyGamma[7/8], 10, 100][[1]] (* Amiram Eldar, Jun 03 2022 *)

Equals (sqrt(2)+1)*Pi/2 - sqrt(2)*arcsinh(1) - 4*log(2) - gamma, where gamma is Euler's constant (A001620). - Amiram Eldar, Jun 03 2022

A355186 Decimal expansion of 2*log(sqrt(2)+1)/Pi.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 23 2022

The average distance between the center of a unit square to a point on its perimeter uniformly chosen by a random direction from the center.

If the point is uniformly chosen at random on the perimeter, then the average is (sqrt(2) + arcsinh(1))/4 (A345653).

			0.56109985233918012713571958893523969280887242462809...

Mark Dalthorp, Escape the Square, Mathematics Magazine, Vol. 87, No. 1 (2014), p. 68.

Cf. A019669, A060294, A091648, A345653.

RealDigits[2*ArcSinh[1]/Pi, 10, 100][[1]]

Equals arcsinh(1)/arcsin(1).

Equals A060294 * A091648 = A091648 / A019669.

A128426 Decimal expansion of the location of a maximum of a Fibonacci Hamiltonian function.

Original entry on oeis.org

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

Offset: 0

Jonathan Vos Post, May 04 2007

The abscissa x of a unique maximum of the f(x) in Theorem 1 of Damanik et al., arising in spectrum of a periodic operator of the one-dimensional Schrodinger equation.

The f(x) at the maximum is A091648.

			0.5395042867...

David Damanik, Mark Embree, Anton Gorodetski, Serguei Tcheremchantsev, The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian, arXiv:0705.0338, 2 May 2007, p. 3.

Decimal expansion of 2*(6 - sqrt(2))/17.

Offset corrected and more digits added by R. J. Mathar, Mar 23 2010

A348669 Decimal expansion of 2sqrt(2)log(1 + sqrt(2))/(3*Pi).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Oct 29 2021

The average length of a random line segment in a unit square defined as follows. A line that is making a random angle with a given edge of the square is chosen, and a random distance of this line from a given vertex of this edge is chosen uniformly between 0 and the distance to the opposite vertex in the square. The segment is then being chosen by picking at random two points between the two intersection points of the line with the perimeter of the square.

			0.26450500700786984551577520129722526936340009096805...

Eric Rosenberg, The expected length of a random line segment in a rectangle, Operations Research Letters, Vol. 32, No. 2 (2004), pp. 99-102.

Cf. A088367, A091505, A091648, A244920.

evalf(sqrt(8/9)*arcsinh(1)/Pi, 120);  # Alois P. Heinz, Oct 29 2021

RealDigits[2*Sqrt[2]*Log[1 + Sqrt[2]]/(3*Pi), 10, 100][[1]]

2*sqrt(2)*log(1 + sqrt(2))/(3*Pi) \\ Michel Marcus, Oct 29 2021

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A354634 Decimal expansion of the negated digamma function at 5/8.

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A354635 Decimal expansion of the negated digamma function at 7/8.

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A355186 Decimal expansion of 2*log(sqrt(2)+1)/Pi.

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A128426 Decimal expansion of the location of a maximum of a Fibonacci Hamiltonian function.

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A348669 Decimal expansion of 2*sqrt(2)*log(1 + sqrt(2))/(3*Pi).

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A348669 Decimal expansion of 2sqrt(2)log(1 + sqrt(2))/(3*Pi).