A164102 -id:A164102 - cp's OEIS frontend

A195055 Decimal expansion of Pi^2/3.

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

Offset: 1

Omar E. Pol, Oct 04 2011

			3.289868133696452872944830333292050378438...

Marc Briane and Gilles Pagès, Théorie de l'Intégration, Vuibert, 2004, 3ème édition, exercice 12.15, p. 256.

George E. Andrews, Partitions with short sequences and mock theta functions, Proceedings of the National Academy of Sciences, Vol. 102, No. 13 (2005), pp. 4666-4671.
Index entries for transcendental numbers

Cf. A002388, A102753, A091476, 2*A013661, A164102.
Cf. A024916 (partial sums of A000203).
Cf. A001008/A002805, A145426.

pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^105*pi^2/3))); // Vincenzo Librandi, Jan 12 2016

RealDigits[Pi^2/3, 10, 105][[1]] (* T. D. Noe, Oct 05 2011 *)

2*zeta(2) \\ Charles R Greathouse IV, Jan 20 2022

sumnumrat(1/x^2,-oo) \\ Charles R Greathouse IV, Jan 20 2022

Equals 3 + A145426.
Equals -Sum_{n>=1} Psi_2(n), where Psi_2 is the tetragamma function. - Istvan Mezo, Oct 25 2012
Equals Integral_{x=0..1} (log(x)/(x - 1))^2 dx. - Jean-François Alcover, Mar 21 2013
Equals Integral_{x=-oo..oo} x^2/sinh(x)^2 dx. - Amiram Eldar, Aug 06 2020
Equals Integral_{x=0..oo} (log(x+1)/x)^2 dx (reference Briane and Pagès). - Bernard Schott, Feb 13 2022
Equals Sum_{n>=1} H(n) * binomial(2*n, n) / (n * 4^n), where H(n) is the n-th harmonic number. - Antonio Graciá Llorente, Apr 04 2025

Extended by T. D. Noe, Oct 05 2011

A182448 Decimal expansion of Pi^2/15.

Original entry on oeis.org

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

Offset: 0

Mats Granvik, Apr 29 2012

			0.65797362673929...

Thomas L. Curtright, Bernoulli Partitions, arXiv:2502.09633 [math.CO], 2025. See p. 3. Mentions this sequence.
László Tóth, Linear combinations of Dirichlet series associated with the Thue-Morse sequence, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 22 (2022), #A98; arXiv preprint, arXiv:2211.13570 [math.NT], 2022.
Index entries for transcendental numbers.

Cf. A000796, A008836, A010060, A013661, A013662, A086463, A191898, A249103.

Cf. A347328, A347329, A347330, A347331.

RealDigits[N[Sum[1/(n + 0)^2 - 1/(n + 1)^2 + 1/(n + 2)^2 - 1/(n + 3)^2 - 4/(n + 4)^2 - 1/(n + 5)^2 + 1/(n + 6)^2 - 1/(n + 7)^2 + 1/(n + 8)^2 + 4/(n + 9)^2, {n, 1, Infinity, 10}], 90]][[1]]
RealDigits[N[Sum[LiouvilleLambda[n]/n^2, {n, 1, Infinity}], 90]][[1]]
RealDigits[Pi^2/15,10,120][[1]] (* Harvey P. Dale, May 28 2017 *)

PARI
```
Pi^2/15 \\ Michel Marcus, Oct 21 2014
```

See Mathematica code.
Equals Gamma(4)*zeta(4)/Pi^2 = zeta(4)/zeta(2) = A013662/A013661 = Product_{p prime} (p^2/(p^2+1)). - Stanislav Sykora, Oct 21 2014
Equals (1/10) * Sum_{n >= 0} (-1)^n*( 1/(n + 1/3)^2 - 1/(n + 2/3)^2 ). - Peter Bala, Oct 31 2019
Equals Sum_{k>=1} A008836(k)/k^2. - Amiram Eldar, Jun 23 2020
Equals (1/10) * Sum_{k>=1} (5*t(k-1) + 3*t(k))/k^2, where t(k) = A010060(k) (Tóth, 2022). - Amiram Eldar, Feb 04 2024
Equals 3/5 + (1/5) * Sum_{n>=1} 1/(n^2*(n+1)^2). - Davide Rotondo, May 28 2025
Equals 1/A082020 = A164102/30 = A195055/5. - Hugo Pfoertner, May 28 2025

Offset corrected and more terms added by Rick L. Shepherd, Jan 08 2014

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

A164103 Decimal expansion of 8*Pi^2/15.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 10 2009

Volume of the 5-dimensional unit sphere.

For all nonnegative integers n, let V_n be the volume of the n-dimensional unit sphere. If n != 5, then V_n < V_5, this constant (see A072345). - Rick L. Shepherd, Feb 23 2014

			Equals 5.2637890139143245967117285332672806...

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 67.

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

RealDigits[8*Pi^2/15, 10, 100][[1]] (* G. C. Greubel, Apr 09 2017 *)

PARI
```
8*Pi^2/15 \\ G. C. Greubel, Apr 09 2017
```

Equals A164104/5 = 4*A164102/15.

A164104 Decimal expansion of 8*Pi^2/3.

Original entry on oeis.org

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

Offset: 2

R. J. Mathar, Aug 10 2009

Surface area of the 5-dimensional unit sphere.

			Equals 26.318945069571622983558642666336...

G. C. Greubel, Table of n, a(n) for n = 2..5000
Eric Weisstein, Hypersphere, Wolfram Mathworld.
Wikipedia, Hypersphere
Index entries for transcendental numbers

Cf. A019692, A019694, A164102, A091925, A164107, A164109.

RealDigits[8*Pi^2/3, 10, 50][[1]] (* G. C. Greubel, Sep 11 2017 *)

PARI
```
8*Pi^2/3 \\ G. C. Greubel, Sep 11 2017
```

Equals 5*A164103 = 10 * A019699 * A019692.

A-number in formula corrected by R. J. Mathar, Aug 12 2010

A164107 Decimal expansion of 16*Pi^3/15.

Original entry on oeis.org

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

Offset: 2

R. J. Mathar, Aug 10 2009

Surface area of the 7-dimensional unit sphere.

			Equals 33.07336179231980818717473607157482154904030780...

G. C. Greubel, Table of n, a(n) for n = 2..5000
Eric Weisstein, Hypersphere, Wolfram Mathworld.
Wikipedia, Hypersphere
Index entries for transcendental numbers

Equals 7*A164106.

Cf. A019692, A019694, A164102, A164104, A091925, A164109.

RealDigits[16*Pi^3/15, 10, 50][[1]] (* G. C. Greubel, Sep 11 2017 *)

16*Pi^3/15 \\ G. C. Greubel, Sep 11 2017

A164081 Floor of 2^(n-1) times the surface area of the unit sphere in 2n-dimensional space.

Original entry on oeis.org

6, 39, 124, 259, 408, 512, 536, 481, 378, 264, 166, 94, 49, 24, 10, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Jonathan Vos Post, Aug 09 2009

nonn

The rounded values of this real sequence is A164082, the ceiling is A164083.
The surface area of n-dimensional sphere of radius r is n*V_n*r^(n-1); see A072478/A072479.
There are only 17 nonzero terms. - G. C. Greubel, Sep 10 2017

			Table of approximate real values before taking integer part.
========================
n (2*Pi)^n / (n-1)!
1 6.28318531 = A019692
2 39.4784176 = 2*A164102
3 124.025107 = 4*A091925
4 259.757576 = 8*A164109
5 408.026246
6 512.740903
7 536.941018
8 481.957131
9 378.528246
10 264.262568
11 166.041068
12 94.8424365
13 49.6593836
14 24.00147
15 10.7718345
16 4.5120955
17 1.77189576
18 0.654891141
19 0.228600133
20 0.075596684
========================

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices, and Groups, 2nd ed., New York: Springer-Verlag, p. 9, 1993.
H. S. M. Coxeter, Regular Polytopes, 3rd ed., New York: Dover, 1973.
D. M. Y. Sommerville, An Introduction to the Geometry of n Dimensions, New York: Dover, p. 136, 1958.

Eric W. Weisstein, Hypersphere.

Cf. A072345, A072346, A072478, A072479, A074457, A122510, A154255.

A164081 := proc(n) (2*Pi)^n/(n-1)! ; floor(%) ; end: seq(A164081(n),n=1..80) ; # R. J. Mathar, Sep 09 2009

Table[Floor[(2*Pi)^n/(n - 1)!], {n, 1, 100}] (* G. C. Greubel, Sep 10 2017 *)

for(n=1,100, print1(floor((2*Pi)^n/(n-1)!), ", ")) \\ G. C. Greubel, Sep 10 2017

a(n) = floor( (2*Pi)^n/(n-1)! ).

Definition corrected by R. J. Mathar, Sep 09 2009

A164082 Rounded value of 2^(n-1) times the surface area of the unit sphere in 2n-dimensional space.

Original entry on oeis.org

6, 39, 124, 260, 408, 513, 537, 482, 379, 264, 166, 95, 50, 24, 11, 5, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Jonathan Vos Post, Aug 09 2009

nonn

The floor of this real sequence is A164081, the ceiling is A164083.
The surface area of the n-dimensional sphere of radius r is n*V_n*r^(n-1); see A072478/ A072479.
There are 18 nonzero terms in this sequence. - G. C. Greubel, Sep 11 2017

			Table of approximate real values before rounding up or down:
========================
n ((2*pi)^n) / (n-1)!
1 6.28318531 = A019692
2 39.4784176 = 2*A164102
3 124.025107 = 4*A091925
4 259.757576 = 8*A164109
5 408.026246
6 512.740903
7 536.941018
8 481.957131
9 378.528246
10 264.262568
11 166.041068
12 94.8424365
13 49.6593836
14 24.00147
15 10.7718345
16 4.5120955
17 1.77189576
18 0.654891141
19 0.228600133
20 0.075596684
========================

Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, p. 9, 1993.
Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.
Sommerville, D. M. Y. An Introduction to the Geometry of n Dimensions. New York: Dover, p. 136, 1958.

Eric W. Weisstein, Hypersphere,

Cf. A072345, A072346, A072478, A072479, A074457, A122510, A154255, A164081, A164083.

A164082 := proc(n) (2*Pi)^n/(n-1)! ; round(%) ; end: seq(A164082(n),n=1..80) ; # R. J. Mathar, Sep 09 2009

Table[Round[(2*Pi)^n/(n - 1)!], {n, 1, 20}] (* G. C. Greubel, Sep 11 2017 *)

for(n=1,20, print1(round((2*Pi)^n/(n-1)!), ", ")) \\ G. C. Greubel, Sep 11 2017

a(n) = round(((2*Pi)^n)/(n-1)!).

Definition corrected by R. J. Mathar, Sep 09 2009

A164083 Ceiling of 2^(n-1) times the surface area of the unit sphere in 2n-dimensional space.

Original entry on oeis.org

7, 40, 125, 260, 409, 513, 537, 482, 379, 265, 167, 95, 50, 25, 11, 5, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Jonathan Vos Post, Aug 09 2009

nonn

The rounded values of this real sequence is A164082, the floor is A164081.

The surface area of n-dimensional sphere of radius r is n*V_n*r^(n-1); see A072478/A072479.

			Table of approximate real values before rounding up.
========================
n ((2*pi)^n) / (n-1)!
1 6.28318531 = A019692
2 39.4784176 = 2*A164102
3 124.025107 = 4*A091925
4 259.757576 = 8*A164109
5 408.026246
6 512.740903
7 536.941018
8 481.957131
9 378.528246
10 264.262568
11 166.041068
12 94.8424365
13 49.6593836
14 24.00147
15 10.7718345
16 4.5120955
17 1.77189576
18 0.654891141
19 0.228600133
20 0.075596684
========================

Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, p. 9, 1993.
Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.
Sommerville, D. M. Y. An Introduction to the Geometry of n Dimensions. New York: Dover, p. 136, 1958.

Eric W. Weisstein, Hypersphere,

Cf. A072345, A072346, A072478, A072479, A074457, A122510, A154255, A164081, A164082.

Table[Ceiling[(2Pi)^n/(n-1)!],{n,60}] (* Harvey P. Dale, Jul 30 2020 *)

a(n) = ceiling(((2*pi)^n)/(n-1)!).

Definition corrected - R. J. Mathar, Sep 09 2009

A195056 Decimal expansion of Pi^2/7.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Oct 04 2011

			1.409943485869908374119212999982307305045...

F. Aubonnet, D. Guinin and B.Joppin, Précis de Mathématiques, Analyse 2, Classes Préparatoires, Premier Cycle Universitaire, Bréal, 1990, Exercice 908, pages 82 and 91-92.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for transcendental numbers

Cf. A000265, A002388, A013661, A019674, A100044, A111003, A164102.

Pi(RealField(128))^2/7; // G. C. Greubel, Jun 02 2021

RealDigits[Pi^2/7, 10, 105][[1]] (* T. D. Noe, Oct 05 2011 *)

PARI
```
Pi^2/7 \\ Michel Marcus, Feb 04 2022
```

numerical_approx(pi^2/7, digits=128) # G. C. Greubel, Jun 02 2021

Equals Sum_{k>=1} A000265(k)/k^3. - Amiram Eldar, Jun 27 2020

Equals Integral_{x=0..1} log(1+x+x^2+x^3+x^4+x^5+x^6)/x dx (Aubonnet). - Bernard Schott, Feb 04 2022

Extended by T. D. Noe, Oct 05 2011

cp's OEIS Frontend

A195055 Decimal expansion of Pi^2/3.

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A182448 Decimal expansion of Pi^2/15.

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A164109 Decimal expansion of Pi^4/3.

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A164103 Decimal expansion of 8*Pi^2/15.

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A164104 Decimal expansion of 8*Pi^2/3.

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A164107 Decimal expansion of 16*Pi^3/15.

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A164081 Floor of 2^(n-1) times the surface area of the unit sphere in 2n-dimensional space.

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