A267315 -id:A267315 - cp's OEIS frontend

A346927 Decimal expansion of the Dirichlet eta function at 10.

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

Offset: 0

Sean A. Irvine, Aug 07 2021

			0.999039507598271565639221845699341831425929649666890...

L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (306).

Michael I. Shamos, Shamos's catalog of the real numbers (2011).
Index entries for transcendental numbers

Cf. A072691, A197070, A267315, A267316, A275703, A275710, A347150, A347059.

RealDigits[DirichletEta[10], 10, 100][[1]] (* Amiram Eldar, Aug 08 2021 *)

-polylog(10, -1) \\ Michel Marcus, Aug 08 2021

Equals 73 * Pi^10 / (2^9 * 3^5 * 5 * 11).
Equals (511/512) * zeta(10).
Equals Sum_{k>=1} (-1)^(k+1) / k^10.
Equals eta(10).

A136038 a(n) = n^6 - n^4.

Original entry on oeis.org

0, 0, 48, 648, 3840, 15000, 45360, 115248, 258048, 524880, 990000, 1756920, 2965248, 4798248, 7491120, 11340000, 16711680, 24054048, 33907248, 46915560, 63840000, 85571640, 113145648, 147756048, 190771200, 243750000, 308458800, 386889048, 481275648

Offset: 0

Rolf Pleisch, Mar 16 2008

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. A013661, A013662, A072691, A267315.

[n^6-n^4: n in [0..30]]; // Vincenzo Librandi, Feb 20 2012

map(n -> n^6 - n^4, [$0..100]); # Robert Israel, Aug 28 2018

a[n_]:=n^6-n^4; a[Range[0,60]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2011 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,48,648,3840,15000,45360},30] (* Harvey P. Dale, May 17 2018 *)

From R. J. Mathar, Feb 06 2010: (Start)
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).
G.f.: 24*x^2*(1+x)*(2*x^2+11*x+2)/(1-x)^7. (End)
From Amiram Eldar, Jan 12 2021: (Start)
Sum_{n>=2} 1/a(n) = 11/4 - Pi^2/6 - Pi^4/90 = 11/4 - A013661 - A013662.
Sum_{n>=2} (-1)^n/a(n) = 7*Pi^4/720 + Pi^2/12 - 7/4 = A267315 + A072691 - 7/4. (End)

A349252 Decimal expansion of Sum_{k>=1} (-1)^k * log(k) / k^4.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Nov 12 2021

First derivative of the Dirichlet eta function at 4.

			0.0334788045785650663859568547887377997137597304057...

Eric Weisstein's World of Mathematics, Dirichlet Eta Function

Cf. A013662, A091812, A210593, A256358, A261506, A267315, A349220.

Flatten[{0, RealDigits[(Pi^4 Log[2] + 630 Zeta'[4])/720, 10, 120][[1]]}]

sumalt(k=1, (-1)^k * log(k) / k^4) \\ Michel Marcus, Nov 12 2021

Equals (Pi^4 * log(2) + 630 * zeta'(4)) / 720.

A246967 Decimal expansion of the real positive solution to eta(x) = x.

Original entry on oeis.org

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

Offset: 0

Michal Paulovic, Sep 08 2014

Fixed point of Dirichlet eta function.

			0.6293340940...

Eric Weisstein's World of Mathematics, Dirichlet Eta Function.
Wikipedia, Dirichlet Eta Function.

Cf. A197070, A267315, A267316.

RealDigits[x /. FindRoot[DirichletEta[x] - x, {x, 0}, WorkingPrecision -> 120], 10, 100] [[1]] (* Amiram Eldar, May 24 2021 *)

solve(n=0,2,(1-2^(1-n))*zeta(n)-n) \\ Edward Jiang, Sep 08 2014

A269481 Continued fraction expansion of the Dirichlet eta function at 4.

Original entry on oeis.org

0, 1, 17, 1, 7, 3, 3, 1, 7, 3, 6, 1, 1, 7, 1, 11, 1, 11, 5, 1, 2, 2, 2, 7, 1, 14, 6, 5, 1, 1, 1, 1, 10, 9, 1, 1, 5, 2, 2, 3, 2, 5, 2, 4, 1, 46, 312, 3, 3, 1, 15, 1, 2, 5, 2, 1, 1, 27, 1, 2, 1, 2, 11, 5, 2, 1, 482, 3, 2, 4, 2, 2, 3, 1, 3, 1, 2, 1, 1, 13, 1, 13, 1, 1, 67, 149, 7, 2, 2, 18, 1, 2, 1, 1, 1, 51, 1, 7, 1, 8

Offset: 0

Ilya Gutkovskiy, Feb 27 2016

Continued fraction of Sum_{k>=1} (-1)^(k - 1)/k^4 = (7*Pi^4)/720 = 0.9470328294972459175765...

			1/1^4 - 1/2^4 + 1/3^4 - 1/4^4 + 1/5^4 - 1/6^4 +... = 1/(1 + 1/(17 + 1/(1 + 1/(7 + 1/(3 + 1/(3 + 1/...)))))).

OEIS Wiki, Euler's alternating zeta function
Eric Weisstein's World of Mathematics, Dirichlet Eta Function
Wikipedia, Dirichlet Eta Function
Index entries for continued fractions for constants

Cf. A013680, A267315.

Mathematica
```
ContinuedFraction[(7 Pi^4)/720, 100]
```

cp's OEIS Frontend

A346927 Decimal expansion of the Dirichlet eta function at 10.

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A136038 a(n) = n^6 - n^4.

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A349252 Decimal expansion of Sum_{k>=1} (-1)^k * log(k) / k^4.

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A246967 Decimal expansion of the real positive solution to eta(x) = x.

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A269481 Continued fraction expansion of the Dirichlet eta function at 4.

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