A164108 -id:A164108 - cp's OEIS frontend

A102753 Decimal expansion of (Pi^2)/2.

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

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Feb 10 2005

Also equals the area under the peak-shaped even function f(x)=x/sinh(x).
Proof: For the upper half of the integral, write f(x) = 2x*exp(-x)/(1-exp(-2x)) = sum_{k=1..infinity} 2x*exp(-(2k-1)x) and integrate term by term from zero to infinity. - Stanislav Sykora, Nov 01 2013
Volume of the 4-dimensional unit sphere; the volume of the n-dimensional unit sphere is Pi^(n/2)/gamma(n/2+1) (see n-ball link and A164103). - Rick L. Shepherd, Jun 22 2017
Pi^2/2 is the squared side-length of a square with diagonal Pi. - Wesley Ivan Hurt, Jan 28 2022

			4.9348022005446793094172454999380755676568497036203953132066746881100\ 224112096026215008867018592761159120129568870115720388....

J. Rivaud, Analyse, Séries, Equations différentielles, Mathématiques Supérieures et Spéciales, Premier Cycle Universitaire, Vuibert, 1981, Exercice 2, p. 135.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Middlesex, England: Penguin Books, 1986, p. 53.

G. C. Greubel, Table of n, a(n) for n = 1..10000
T. Amdeberhan, L. Medina and V. H. Moll, The integrals in Gradshteyn and Ryzhik. Part 5: Some trigonometric integrals, equation 2.39, arXiv:0705.2379 [math.CA], 2007.
Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, El. J. Combin. Numb. Th. 6 (2006) # A27
Eric Weisstein's World of Mathematics, Hypersphere.
Eric Weisstein's World of Mathematics, Trigamma Function.
Wikipedia, Hypersphere.
Wikipedia, Volume of an n-ball.
Index entries for transcendental numbers

Cf. A002388, A000796, A019699, A026424, A164103, A164105, A164106, A164108, A248359, A276023.

RealDigits[Pi^2/2, 10, 111][[1]] (* Robert G. Wilson v, Dec 15 2005 *)

PARI
```
Pi^2/2 \\ Michel Marcus, Sep 04 2015
```

Equals psi_1(1/2), where psi_1(x) is the second logarithmic derivative of GAMMA(x).
Equals the volume of revolution of the sine or cosine curve for one half period, Integral_{0,Pi} Sin(x)^2 dx. - Robert G. Wilson v, Dec 15 2005
Equals Sum_{k >=1} 4^k/(k^2*binomial(2*k,k)) [Amdeberhan, Sprugnoli]. - R. J. Mathar, Sep 28 2007
Equals 4*Sum_{k >=1} 1/(2k-1)^2 [Wells].
From  Peter Bala, Nov 05 2019: (Start)
Pi^2/2 = Integral_{x = 0..inf} cosh(x)*x^2/sinh(x)^2 dx.
Pi^2/2 = 5*sum_{k >= 0} binomial(2*k,k)(-1/16)^k*1/(2*k+1)^2.
Pi^2/2 = 10*Integral_{x = 0..1/2}  1/x*log(x + sqrt(1 + x^2)) dx. (End)
Pi^2/20 = 0.1 * Pi^2/2 = Sum_{k>=1} 1/A026424(k)^2. - Amiram Eldar, Aug 17 2020
Conjecture: Pi^2/2 = Sum_{n = -oo..oo} ( cos(Pi*sqrt(n^2+1)) - cos(Pi*n) ) (using the Eisenstein summation convention). - Peter Bala, Oct 08 2021
Pi^2/2 = Integral_{x = -oo..oo} x/sinh(x) dx (see Rivaud reference). - Bernard Schott, Jan 28 2022

A164109 Decimal expansion of Pi^4/3.

Original entry on oeis.org

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

Offset: 2

R. J. Mathar, Aug 10 2009

Surface area of the 8-dimensional unit sphere.

			32.4696970113341457454801108962350370832425285...

G. C. Greubel, Table of n, a(n) for n = 2..5000
Eric Weisstein's World of Mathematics, Hypersphere.
Wikipedia, Hypersphere.
Ce Xu and Jianqiang Zhao, Sun's Three Conjectures on Apéry-like Sums Involving Harmonic Numbers, arXiv:2203.04184 [math.NT], 2022.
Index entries for transcendental numbers.

Cf. A019692, A019694, A164102, A164104, A091925, A164107.

Cf. A001008, A002805.

RealDigits[Pi^4/3,10,120][[1]] (* Harvey P. Dale, Jul 31 2014 *)

PARI
```
Pi^4/3 \\ G. C. Greubel, Sep 11 2017
```

Equals 8*A164108 = A019670*A091925 = A092425/3.

Pi^4/30 = Sum_{k>=1} H(k)*H(k+1)/(k+1)^2, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Aug 19 2025

A164106 Decimal expansion of 16*Pi^3/105.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 10 2009

Volume of the 7-dimensional unit sphere.

			Equals 4.72476597033140116959639086736783164986290111...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein, Hypersphere, Wolfram Mathworld.
Wikipedia, Hypersphere
Index entries for transcendental numbers

Cf. A000796, A019699, A102753, A164103, A164105, A164108.

RealDigits[16*Pi^3/105, 10, 100][[1]] (* G. C. Greubel, Apr 09 2017 *)

16*Pi^3/105 \\ G. C. Greubel, Apr 09 2017

Equals 16*A091925/105 = A164107/7 .

A164105 Decimal expansion of Pi^3/6.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 10 2009

Volume of the 6-dimensional unit sphere.

			Equals 5.1677127800499700292460525111835658670375480943...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein, Hypersphere, Wolfram Mathworld.
Wikipedia, Hypersphere
Index entries for transcendental numbers

Cf. A000796, A019699, A102753, A164103, A164106, A164108.

RealDigits[Pi^3/6, 10, 100][[1]] (* G. C. Greubel, Apr 09 2017 *)

PARI
```
Pi^3/6 \\ G. C. Greubel, Apr 09 2017
```

Equals A091925/6 = A019670*A102753.

A276023 Decimal expansion of 32*Pi^4/945.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Aug 16 2016

Volume of the 9-dimensional unit sphere.

More generally, the ordinary generation function for the volume of the n-dimensional unit sphere is exp(Pi*x^2)*(erf(sqrt(Pi)*x) + 1) = 1 + 2*x + Pi*x^2 + (4*Pi/3)*x^3 + (Pi^2/2)*x^4 + ...

			3.2985089027387068693821065037445117...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Ilya Gutkovskiy, Volume of the n-dimensional unit sphere
Eric Weisstein's World of Mathematics, Hypersphere
Index entries for transcendental numbers

Cf. A021949, A092425.

Cf. similar sequences of the volume of the n-dimensional unit sphere: A000796 (n = 2), 10*A019699 (n = 3), A102753 (n = 4), A164103 (n = 5), A164105 (n = 6), A164106 (n = 7), A164108 (n = 8).

Mathematica
```
RealDigits[(32 Pi^4)/945, 10, 120][[1]]
```

(32*Pi^4)/945 \\ G. C. Greubel, Apr 09 2017

Equals 32*A092425*A021949.

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A102753 Decimal expansion of (Pi^2)/2.

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A164109 Decimal expansion of Pi^4/3.

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A164106 Decimal expansion of 16*Pi^3/105.

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A164105 Decimal expansion of Pi^3/6.

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A276023 Decimal expansion of 32*Pi^4/945.

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