A358517 -id:A358517 - cp's OEIS frontend

A358516 Decimal expansion of Sum_{k >= 1} (-1)^(k+1)1/((k+2)(k+3)).

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

Offset: 0

Claude H. R. Dequatre, Nov 20 2022

			0.0529610277865572855011309095830198028176669353...

Michael I. Shamos, A catalog of the real numbers (2011) p.100.

Cf. A002378, A242023, A242024, A358517, A358519.

Cf. A002162.

Join[{0}, RealDigits[2*Log[2] - 4/3, 10, 120][[1]]] (* Amiram Eldar, Nov 21 2022 *)

PARI
```
2*log(2) - 4/3
```

Equals Sum_{k >= 1} (-1)^(k+1)*/((k+2)*(k+3)) = A016627 -4/3.
Equals 2*log(2) - 4/3 = Sum_{k >= 2} 1/(4*k^3 - k) = Sum_{k >= 1} (zeta(2*k + 1) - 1)/(4^k). [from the Shamos reference]
Equals Sum_{k >= 1} 1/((2^k)*(4*k + 12)). [from the Shamos reference]
Equals Sum_{k>=3} (-1)^(k+1)/A002378(k). - Amiram Eldar, Nov 21 2022

Missing terms 6, 0 inserted after a(74) by Georg Fischer, Feb 07 2025

A358519 Decimal expansion of Sum_{k >= 1} (-1)^(k+1)/(k^2 + 4*k - 1).

Original entry on oeis.org

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

Offset: 0

Claude H. R. Dequatre, Nov 20 2022

			0.189957907718062725271908409063636655279566491...

Michael I. Shamos, A catalog of the real numbers (2011), p. 253.

Cf. A028875, A242023, A242024, A358516, A358517.

Cf. A002162.

RealDigits[(2*Sqrt[5]*Pi*Csc[Sqrt[5]*Pi] - 17)/20, 10, 120][[1]] (* Amiram Eldar, Nov 21 2022 *)

(Pi/(2*sqrt(5)))*1/sin(Pi*sqrt(5)) - 17/20

Equals Sum_{k >=1} (-1)^(k+1)/(k^2 + 4*k - 1).
Equals (Pi/(2*sqrt(5)))*csc(Pi*sqrt(5)) - 17/20 = Sum_{k>=3} (-1)^(k+1)/(k^2-5). [from the Shamos reference]
Equals Sum_{k>=3} (-1)^(k+1)/A028875(k). - Amiram Eldar, Nov 21 2022

A365524 Decimal expansion of 4*log(2) - 5/2.

Original entry on oeis.org

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

Offset: 0

Claude H. R. Dequatre, Sep 08 2023

This sequence is also the decimal expansion of Sum_{k>=1} (-1)^(k+1)*f(k), where f(k) = (3*k^2 - k)/(k^2 + k) is the ratio between the k-th pentagonal and triangular numbers.

			0.27258872223978123766892848583270627230200053744102...

Michael Ian Shamos, A catalog of the real numbers (2011), p. 330.
Wikipedia, Polygonal number.
Index entries for transcendental numbers

Cf. A002162.
Cf. A000217, A000326, A358517.
Essentially the same as A016639 and A358517.

RealDigits[4*Log[2] - 5/2, 10 , 100][[1]] (* Amiram Eldar, Sep 08 2023 *)

PARI
```
4*log(2)-5/2
```

Equals Sum_{k>=1} 1/(2^k*(k + 2)) [Shamos].

Equals Sum_{k>=1} (-1)^(k+1)*(3*k^2 - k)/(k^2 + k).

A369882 Decimal expansion of log(2) - 1/4.

Original entry on oeis.org

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

Offset: 0

Claude H. R. Dequatre, Feb 04 2024

			0.443147180559945309417232121458176568075500134...

Michael Ian Shamos, A catalog of the real numbers (2011), p. 453.
Index entries for transcendental numbers

Cf. A358517, A365524.

Essentially the same as A002162 and A187832.

RealDigits[Log[2] - 1/4, 10, 120][[1]] (* Amiram Eldar, Feb 04 2024 *)

PARI
```
log(2)-1/4
```

Equals Sum_{k >= 2} (-1)^k*k/(k^2 - 1) [Shamos].

Equals Sum_{k >= 1} (-1)^(k+1)*(k + 1)/(k*(k + 2)).

cp's OEIS Frontend

A358516 Decimal expansion of Sum_{k >= 1} (-1)^(k+1)*1/((k+2)*(k+3)).

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A358519 Decimal expansion of Sum_{k >= 1} (-1)^(k+1)/(k^2 + 4*k - 1).

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A365524 Decimal expansion of 4*log(2) - 5/2.

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A369882 Decimal expansion of log(2) - 1/4.

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A358516 Decimal expansion of Sum_{k >= 1} (-1)^(k+1)1/((k+2)(k+3)).