A300731 -id:A300731 - 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

A300690 Decimal expansion of sqrt(Pi^2/8 - 1).

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Mar 11 2018

Also the total harmonic distortion (THD) of a square wave, see formula (11) in the Blagouchine & Moreau link.

			0.4834258476086790990137326370639317022328017276651459...

I. V. Blagouchine and E. Moreau, Analytic Method for the Computation of the Total Harmonic Distortion by the Cauchy Method of Residues. IEEE Trans. Commun., vol. 59, no. 9, pp. 2478-2491, 2011. PDF file.
Index entries for transcendental numbers

Cf. A002388, A111003, A300713, A300714, A300727, A300731.

MATLAB
```
format long; sqrt(pi^2/8-1)
```
Maple
```
evalf(sqrt((1/8)*Pi^2-1), 120)
```

RealDigits[Sqrt[Pi^2/8 - 1], 10, 120][[1]]

default(realprecision, 120); sqrt(Pi^2/8-1)

A300713 Decimal expansion of sqrt(Pi^2/6 - 1) = sqrt(zeta(2) - 1).

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Mar 11 2018

Also the total harmonic distortion (THD) of a sawtooth signal, see formula (15) in the Blagouchine & Moreau link.

			0.8030778709740584281843212446690348231898910996409661...

I. V. Blagouchine and E. Moreau, Analytic Method for the Computation of the Total Harmonic Distortion by the Cauchy Method of Residues. IEEE Trans. Commun., vol. 59, no. 9, pp. 2478-2491, 2011. PDF file.
Index entries for transcendental numbers

Cf. A013661, A300690, A300714, A300727, A300731.

MATLAB
```
format long; sqrt(pi^2/6-1)
```
Maple
```
evalf(sqrt((1/6)*Pi^2-1), 120)
```

RealDigits[Sqrt[Pi^2/6 - 1], 10, 120][[1]]

default(realprecision, 120); sqrt(Pi^2/6-1)

Equals sqrt(A013661 - 1).

cp's OEIS Frontend

A300707 Decimal expansion of Pi^4/96.

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A300690 Decimal expansion of sqrt(Pi^2/8 - 1).

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A300713 Decimal expansion of sqrt(Pi^2/6 - 1) = sqrt(zeta(2) - 1).

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Author

Keywords

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Crossrefs

Programs

MATLAB

Maple

Mathematica

PARI

Formula