A300709 -id:A300709 - cp's OEIS frontend

A300707 Decimal expansion of Pi^4/96.

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

Offset: 1

Iaroslav V. Blagouchine, Mar 11 2018

Also the sum of the series Sum_{n>=0} (1/(2n+1)^4), whose value is obtained from zeta(4) given by L. Euler in 1735: Sum_{n>=0} (2n+1)^(-s) = (1-2^(-s))*zeta(s).

For the partial sums of this series see A120269/A128493. - Wolfdieter Lang, Sep 02 2019

			1.0146780316041920545462534655073449088513290174238064...

Eric Weisstein's World of Mathematics, Dirichlet Lambda Function. See (6).
Index entries for transcendental numbers

Cf. A013662, A092425, A111003, A120269, A128493, A300709, A300710, A300731.

MATLAB
```
format long; pi^4/96
```
Maple
```
evalf((1/96)*Pi^4, 120)
```
Mathematica
```
RealDigits[Pi^4/96, 10, 120][[1]]
```
PARI
```
default(realprecision, 120); Pi^4/96
```

Equals A092425/96. - Omar E. Pol, Mar 11 2018
Equals (15/16)*zeta(4) = (15/16)*A013662. - Wolfdieter Lang, Sep 02 2019
Equals Sum_{k>=1} 1/(2*k-1)^4. - Sean A. Irvine, Mar 25 2025
Equals lambda(4), where lambda is the Dirichlet lambda function. - Michel Marcus, Aug 15 2025

A016758 a(n) = (2*n+1)^6.

Original entry on oeis.org

1, 729, 15625, 117649, 531441, 1771561, 4826809, 11390625, 24137569, 47045881, 85766121, 148035889, 244140625, 387420489, 594823321, 887503681, 1291467969, 1838265625, 2565726409, 3518743761, 4750104241, 6321363049, 8303765625, 10779215329, 13841287201, 17596287801

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A300709.

[(2*n+1)^6: n in [0..30]]; // Vincenzo Librandi, Sep 07 2011

(2*Range[0,20]+1)^6 (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{1, 729,15625,117649,531441,1771561,4826809},20] (* Harvey P. Dale, Dec 26 2012 *)

vector(30, n, n--; (2*n+1)^6) \\ G. C. Greubel, Sep 15 2018

a(0)=1, a(1)=729, a(2)=15625, a(3)=117649, a(4)=531441, a(5)=1771561, a(6)=4826809, a(n) = 7*a(n-1) -21*a(n-2) +35*a(n-3) -35*a(n-4) + 21*a(n-5) -7*a(n-6) +a(n-7). - Harvey P. Dale, Dec 26 2012
G.f.: (1 +722*x +10543*x^2 +23548*x^3 +10543*x^4 +722*x^5 +x^6)/(1-x)^7 . - R. J. Mathar, Jul 07 2017
Sum_{n>=0} 1/a(n) = Pi^6/960 (A300709). - Amiram Eldar, Oct 10 2020
From Amiram Eldar, Jan 28 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = cosh(Pi/2)*(cos(sqrt(3)*Pi/2) + cosh(Pi/2))/2.
Product_{n>=1} (1 - 1/a(n)) = Pi*cosh(sqrt(3)*Pi/2)/24. (End)

A300710 Decimal expansion of 17*Pi^8/161280.

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Mar 11 2018

Also the sum of the series Sum_{n>=0} (1/(2n+1)^8), whose value is obtained from zeta(8) given by L. Euler in 1735: Sum_{n>=0} (2n+1)^(-s)=(1-2^(-s))*zeta(s).

			1.0001551790252961193029872492957280415665429750613740...

Jason Bard, Table of n, a(n) for n = 1..10000
Index entries for transcendental numbers
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 21.

Cf. A092736, A111003, A300707, A300709.

MATLAB
```
format long; (17/161280)*pi^8
```
Maple
```
evalf((17/161280)*Pi^8, 120);
```

RealDigits[(17/161280)*Pi^8, 10, 120][[1]]

default(realprecision, 120); (17/161280)*Pi^8

Equals 17*A092736/161280. - Omar E. Pol, Mar 11 2018
From Artur Jasinski, Jun 24 2025: (Start)
Equals DirichletL(2,1,8).
Equals DirichletL(4,1,8).
Equals DirichletL(8,1,8).
Equals DirichletL(16,1,8). (End)
Equals 255*Zeta(8)/256. - Jason Bard, Aug 21 2025

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A300707 Decimal expansion of Pi^4/96.

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A016758 a(n) = (2*n+1)^6.

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A300710 Decimal expansion of 17*Pi^8/161280.

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MATLAB

Maple

Mathematica

PARI

Formula