A101293 -id:A101293 - cp's OEIS frontend

A063374 a(n) = Fibonacci(n!).

1, 1, 1, 8, 46368, 5358359254990966640871840, 1323171012053243520828784042795469593341319770463238313551473338336502410952765153371119398122747569819754164672344667591018783803781288766524146031040

Offset: 0

Jason Earls, Jul 23 2001

nonn

Cf. A000045, A000142, A101293.

a:= n-> (<<0|1>, <1|1>>^(n!))[1, 2]:
seq(a(n), n=0..6);  # Alois P. Heinz, Jun 10 2018

Array[Fibonacci[ #! ]&,7] (* Vladimir Joseph Stephan Orlovsky, Nov 01 2009 *)

PARI
```
for(n=0,6,print(fibonacci(n!)))
```

[fibonacci(factorial(n)) for n in range(0, 9)] # Zerinvary Lajos, Nov 30 2009

a(n) = A000045(A000142(n)).

A371323 Decimal expansion of Sum_{k>=1} 1/(2^k * Lucas(k!)).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Mar 19 2024

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

			0.59027838058250762481004953440302226140463948387293...

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. A000032, A000142, A101293, A343202, A371322, A371324, A371325.

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))))

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)))

A371137 Decimal expansion of Sum_{k>=1} 1/Lucas(k!).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 12 2024

Nyblom (2000) proved that this constant is transcendental.

			1.38889853376456644140523703662326084973849454043352...

M. A. Nyblom, A theorem on transcendence of infinite series, The Rocky Mountain Journal of Mathematics, Vol. 30, No. 3 (2000), pp. 1111-1120; alternative link.
Index entries for transcendental numbers.

Cf. A000032, A000142, A101293, A343202, A371136.

RealDigits[Sum[1/LucasL[k!], {k, 1, 10}], 10, 120][[1]]

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

Equals Sum_{k>=1} 1/A101293(k).

cp's OEIS Frontend

A063374 a(n) = Fibonacci(n!).

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A371323 Decimal expansion of Sum_{k>=1} 1/(2^k * Lucas(k!)).

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

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A371137 Decimal expansion of Sum_{k>=1} 1/Lucas(k!).

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