A164102 -id:A164102 - cp's OEIS frontend

A377588 Decimal expansion of 7zeta(3)/(2Pi^2) - log(2) + 1/2.

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

Offset: 0

Stefano Spezia, Nov 02 2024

			0.233131218257560481506289305137990304982506635269...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, pp. 43-44.

Cf. A002117, A002162, A094642, A164102, A244009, A377589, A377592.

RealDigits[7Zeta[3]/(2Pi^2)-Log[2]+1/2,10,100][[1]]

Equals Sum_{k>=1} zeta(2*k)/((k + 1)*2^(2*k)) (see Finch).

A377589 Decimal expansion of 9zeta(3)/(2Pi^2) - log(2) + 1/3.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Nov 02 2024

			0.18825837982446689796062876035594274490384190278...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, pp. 43-44.

Cf. A002117, A002162, A094642, A164102, A244009, A377588, A377592.

RealDigits[9Zeta[3]/(2Pi^2)-Log[2]+1/3,10,100][[1]]

Equals Sum_{k>=1} zeta(2*k)/((2*k + 3)*2^(2*k-1)) (see Finch).

A233699 Ideal rectangle side length for packing squares with side 1/n.

Original entry on oeis.org

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

Offset: 0

John W. Nicholson, Dec 15 2013

With one side s_1 = 1/2+1/3 = 5/6, and with area A = s_1*s_2 = sum(n=2,infinity, 1/n^2) = Pi^2/6 - 1 = A013661 - 1, the second side, s_2, can be solved.

The current packing record holder is Marc Paulhus, who developed a packing algorithm (see Link).

			0.77392088021787172376689819997523022706273988144815812528266987524400896448...

G. C. Greubel, Table of n, a(n) for n = 0..10000
M. M. Paulhus, An Algorithm for Packing Squares, Journal of Combinatorial Theory,1998, A,82(2), pages 147-157.
Pegg Jr, Ed., Wolfram Demonstrations Project, Packing Squares with Side 1/n
Wikipedia, Packing Squares with Side 1/n

Essentially the same as A164102.

C := ComplexField(); (Pi(C)^2-6)/5 // G. C. Greubel, Jan 26 2018

Mathematica

RealDigits[(Pi^2-6)/5,10,120][[1]] (* Harvey P. Dale, Aug 21 2017 *)

PARI

(Pi^2-6)/5;

Formula

Equals (Pi^2-6)/5 = A164102/10 - 6/5.

A261791 The integer part of the surface area of the 4-dimensional sphere of radius n.

Original entry on oeis.org

19, 157, 532, 1263, 2467, 4263, 6770, 10106, 14389, 19739, 26272, 34109, 43367, 54164, 66619, 80851, 96978, 115119, 135391, 157913, 182804, 210183, 240166, 272874, 308425, 346936, 388526, 433315, 481419, 532958

Offset: 1

Ilya Gutkovskiy, Sep 01 2015

nonn

2*Pi^2*n^3 - surface area of the 4-dimensional sphere of radius n.

Eric Weisstein's World of Mathematics, Hypersphere
Eric Weisstein's World of Mathematics, Floor Function
Wikipedia, N-sphere

Cf. A038130, A000578, A164102 (2*Pi^2), A210519, A066644.

Mathematica
```
Table[Floor[2 Pi^2 n^3], {n, 1, 30}]
```

a(n) = floor(2*Pi^2*n^3) \\ Charles R Greathouse IV, Sep 18 2015

a(n) = floor(2*Pi^2*n^3).

a(n) = floor(2*Pi^2*A000578(n)).

A325629 Floor of number of n-dimensional degrees in an n-sphere.

Original entry on oeis.org

2, 360, 41252, 3712766, 283634468, 19145326633, 1170076174384, 65816784809141, 3447793362911604, 16969079580805447, 7901760333122072321, 350023289756266797348, 14816864219294689084225

Offset: 0

Eliora Ben-Gurion, Sep 07 2019

nonn

Only the 0th and 1st terms of this sequence are exact values of n-degrees in an n-sphere, by definition. The 0-sphere, being 2 disconnected points at the ends of a segment, is trivial.
The number of degrees, minutes, seconds in an n-sphere is designed to approximate the size of an n-cube, m^n units in size, as m becomes increasingly small, observed from the center of the sphere. This makes a degree Pi/180 of a radian, a square degree (Pi/180)^2 of a steradian, a cubic degree (Pi/180)^3 of a 3-radian, etc.
The sequence has a maximum value at n = 20626 with a value of 1.3610489172...*10^4479, too large to be written here. I conjecture that the peak value of the function analytically is somewhere near 64800/Pi = 20626.48062...
At n = 56058 the sequence has a value of 281 (actual number 281.4089), meaning the 56058-dimensional sphere has less than 360 degrees. At n = 56070, the function has a value of 0.6978855, turning the rest of the sequence into a string of zeros.
An "N-sphere" is located in an N+1-dimensional space, 1-sphere being a circle, 2-sphere being an ordinary sphere, and so on.
From Jon E. Schoenfield, Sep 07 2019: (Start)
The maximum value of the continuous function is 1.361052727810610001492173640278424460497...*10^4479 and it occurs at 20626.48061662940750570152124725484602696... which is close to 64800/Pi, but it's actually 64799.99997461521504462375443773494034381.../Pi. That numerator appears to be 64800 - z/64800 + (27/10) * z^2 / 64800^3 - ... where z = zeta(2) = Pi^2 / 6. (End)

			Number of cubic degrees in a 3-sphere:
Surface area of a 3-sphere: 2*Pi^((3+1)/2) / ((3+1)/2 - 1)! = 2*Pi^2 / (2-1)! = 2*Pi^2.
Cubic degrees: 2*Pi^2 * (180/Pi)^3 = 11664000 / Pi = 3712766.512...
Number of tesseractic degrees in a 4-sphere:
Surface area of a 4-sphere: 2*Pi^((4+1)/2) / Gamma(5/2) = 2*Pi^(5/2) / (3*Pi^(1/2)/4) = 8*Pi^2/3.
Tesseractic degrees: 8*Pi^2/3 * (180/Pi)^4 = 2799360000 / Pi^2 = 283634468.641...

Eric Weisstein's World of Mathematics, Solid Angle.
Wikipedia, N-sphere
Wikipedia, Particular values of the gamma function

Surface area of k-dimensional sphere for k=2..8: A019692, A019694, A164102, A164104, A091925, A164107, A164109.

Cf. A125560.

a(n) = floor((2*Pi^((n+1)/2)/((n+1)/2-1)!)/(Pi/180)^n).
a(n) = floor((2*Pi^((n+1)/2)/(Gamma((n+1)/2)))/(Pi/180)^n).
a(n) = floor(2^(2n+1)*45^n*Pi^((n+1)/2-n)/((n+1)/2-1)!).
a(n) = floor(2^(2n+1)*45^n*Pi^((n+1)/2-n)/(Gamma((n+1)/2))).

A377645 Decimal expansion of 2Pi^2log(2) - 7*zeta(3).

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Nov 03 2024

			5.26777860559707309189679117732804220724465404999...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.5, p. 46.

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 305.

Cf. A002117, A002162, A002388, A016627, A164102.

RealDigits[2Pi^2Log[2]-7Zeta[3],10,100][[1]]

Equals -4*Integral_{x=0..Pi} x*log(sin(x/2)) dx = 8*Integral_{x=0..1} arcsin(x)^2/x dx = 8*Integral_{x=0..Pi/2} x^2*cot(x) dx (see Finch).

Equals Integral_{x=0..Pi} x^2*sin(x)/(1 - cos(x)) dx (see Shamos).

A379391 Decimal expansion of sqrt(Pi^2 - 4)/(2*Pi^2).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Dec 22 2024

			0.1227367657057171588878865032271413191796583881...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.4.2, p. 494.

Cf. A002388, A164102.

RealDigits[Sqrt[Pi^2-4]/(2Pi^2),10,100][[1]]

cp's OEIS Frontend

A377588 Decimal expansion of 7*zeta(3)/(2*Pi^2) - log(2) + 1/2.

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A377589 Decimal expansion of 9*zeta(3)/(2*Pi^2) - log(2) + 1/3.

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A233699 Ideal rectangle side length for packing squares with side 1/n.

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Magma

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A261791 The integer part of the surface area of the 4-dimensional sphere of radius n.

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PARI

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A325629 Floor of number of n-dimensional degrees in an n-sphere.

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A377645 Decimal expansion of 2*Pi^2*log(2) - 7*zeta(3).

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A379391 Decimal expansion of sqrt(Pi^2 - 4)/(2*Pi^2).

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A377588 Decimal expansion of 7zeta(3)/(2Pi^2) - log(2) + 1/2.

A377589 Decimal expansion of 9zeta(3)/(2Pi^2) - log(2) + 1/3.

A377645 Decimal expansion of 2Pi^2log(2) - 7*zeta(3).