A013680 -id:A013680 - cp's OEIS frontend

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

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]
```

A343244 Position of the first occurrence of an element in the continued fraction of zeta(n) which is larger than the second element.

Original entry on oeis.org

5, 4, 8, 14, 10, 63, 120, 79, 1270, 779, 1749, 3410, 13668, 17704, 20909, 175782, 127426

Offset: 2

Amiram Eldar, Apr 08 2021

a(20) = 111604.

The corresponding values of the a(n)-th elements are 4, 18, 183, 32, 61, 9283, 462, 1483, 3530, 3484, 10812, 8954, ...

			The continued fraction of zeta(3) is [1; 4, 1, 18, 1, 1, ...]. The first element which is larger than 4 is 18 whose position is 4. Therefore, a(3) = 4.

Cf. A013697.

Cf. A013679, A013631, A013680, A013681, A013682, A013683, A013684, A013685.

a[n_] := Module[{c = ContinuedFraction[Zeta[n], 10000]}, FirstPosition[c, _?(# > c[[2]] &)][[1]]]; Array[a, 10, 2]

cp's OEIS Frontend

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

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