A371323 -id:A371323 - cp's OEIS frontend

A371322 Decimal expansion of Sum_{k>=1} 1/(2^k * Fibonacci(k!)).

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

Offset: 0

Amiram Eldar, Mar 19 2024

The transcendence of this constant was proved by Nyblom (2001).

			0.76562634791235334713595583744001463296000771662912...

M. A. Nyblom, A Theorem on Transcendence of Infinite Series II, Journal of Number Theory, Vol. 91, No. 1 (2001), pp. 71-80.
Index entries for transcendental numbers.

Cf. A000045, A000142, A063374, A343202, A371323, A371324, A371325.

RealDigits[Sum[1/(2^k * Fibonacci[k!]), {k, 1, 10}], 10, 120][[1]]

PARI
```
suminf(k = 1, 1/(2^k * fibonacci(k!)))
```

A371324 Decimal expansion of Sum_{k>=1} (-1)^(k+1)/(2^k * Fibonacci(k!)).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Mar 19 2024

The transcendence of this constant was proved by Nyblom (2004).

			0.26562365208764665286404417422400359086200909689137...

M. A. Nyblom, An extension of a result of Sierpiński, Journal of Number Theory, Vol. 105, No. 1 (2004), pp. 49-59.
Index entries for transcendental numbers.

Cf. A000045, A000142, A063374, A343202, A371322, A371323, A371325.

RealDigits[-Sum[(-1/2)^k/Fibonacci[k!], {k, 1, 10}], 10, 120][[1]]

PARI
```
suminf(k = 1, -(-1/2)^k/fibonacci(k!))
```

A371325 Decimal expansion of Sum_{k>=1} (-1)^(k+1)/(2^k * Lucas(k!)).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Mar 19 2024

The transcendence of this constant was proved by Nyblom (2004).

			0.42361050830638126407883935970217415533672669926872...

M. A. Nyblom, An extension of a result of Sierpiński, Journal of Number Theory, Vol. 105, No. 1 (2004), pp. 49-59.
Index entries for transcendental numbers.

Cf. A000032, A000142, A101293, A343202, A371322, A371323, A371324.

RealDigits[-Sum[(-1/2)^k/LucasL[k!], {k, 1, 10}], 10, 120][[1]]

suminf(k = 1, -(-1/2)^k/(fibonacci(k!-1)+fibonacci(k!+1)))

cp's OEIS Frontend

A371322 Decimal expansion of Sum_{k>=1} 1/(2^k * Fibonacci(k!)).

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A371324 Decimal expansion of Sum_{k>=1} (-1)^(k+1)/(2^k * Fibonacci(k!)).

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A371325 Decimal expansion of Sum_{k>=1} (-1)^(k+1)/(2^k * Lucas(k!)).

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Mathematica

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