A063374 -id:A063374 - cp's OEIS frontend

A101293 a(n) = Lucas(n!).

1, 1, 3, 18, 103682, 11981655542024930675232002

Offset: 0

Parthasarathy Nambi, Dec 21 2004

The next term is too large to include.

The next term has 151 digits. - Harvey P. Dale, Jan 04 2018

			L(3!) = 18.
L(4!) = 103682.

Alois P. Heinz, Table of n, a(n) for n = 0..6

Cf. A000032, A000142, A063374.

a:= n-> (<<1|1>, <1|0>>^n!. <<2, -1>>)[1, 1]:
seq(a(n), n=0..6);  # Alois P. Heinz, Jul 15 2025

LucasL[Range[0,5]!] (* Harvey P. Dale, Jan 04 2018 *)

a(n) = A000032(A000142(n)).

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

Original entry on oeis.org

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!))
```

A261626 a(n) = Fibonacci(n!) - Fibonacci(n)!.

Original entry on oeis.org

0, 0, 0, 6, 46362, 5358359254990966640871720

Offset: 0

Altug Alkan, Sep 14 2015

nonn

			For n = 1, a(n) = Fibonacci(n!) - Fibonacci(n)! = 1 - 1 = 0.

Cf. A000045, A060001, A063374.

a(n) = fibonacci(n!) - fibonacci(n)!;
vector(6, n, a(n-1))

a(n) = A063374(n) - A060001(n).

log log a(n) ~ n log n. - Charles R Greathouse IV, Sep 14 2015

A262776 a(n) = Fibonacci(n!) mod Fibonacci(n)!.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 20160, 1098377280, 10712200669548618240, 157910199555786679826546221836620444160, 12162675222629942931022379230724715707339402614012620710827200735689241600

Offset: 0

Altug Alkan, Oct 01 2015

Inspired by A261626.

Is there a possibility of observing that a(n) = 0 for n > 5?

			a(0) = Fibonacci(0!) mod Fibonacci(0)! = 1 mod 1 = 0.
a(1) = Fibonacci(1!) mod Fibonacci(1)! = 1 mod 1 = 0.
a(2) = Fibonacci(2!) mod Fibonacci(2)! = 1 mod 1 = 0.
a(3) = Fibonacci(3!) mod Fibonacci(3)! = 8 mod 2 = 0.
a(4) = Fibonacci(4!) mod Fibonacci(4)! = 46368 mod 6 = 0.

Alois P. Heinz, Table of n, a(n) for n = 0..14

Cf. A000045, A060001, A063374, A261626.

[Fibonacci(Factorial(n)) mod Factorial(Fibonacci(n)): n in [0..10]]; // Vincenzo Librandi, Oct 01 2015

Table[Mod[Fibonacci[n!], Fibonacci[n]!], {n, 0, 9}] (* Michael De Vlieger, Oct 01 2015 *)

a(n) = fibonacci(n!) % fibonacci(n)!;
vector(10, n, a(n-1))

from gmpy2 import fac, fib
def A262776(n):
    if n < 2:
        return 0
    a, b, m = 0, 1, fac(fib(n))
    for i in range(fac(n)-1):
        b, a = (b+a) % m, b
    return int(b) # Chai Wah Wu, Oct 03 2015

a(n) = A063374(n) mod A060001(n), for n > 0.

a(10) from Alois P. Heinz, Oct 01 2015

A371136 Decimal expansion of Sum_{k>=1} 1/Fibonacci(k!).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 12 2024

Nyblom (2000) proved that this constant is transcendental.

			2.12502156659765355417529349235237991793625797423002...

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. A000045, A000142, A063374, A343202, A371137.

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

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

Equals Sum_{k>=1} 1/A063374(k).

cp's OEIS Frontend

A101293 a(n) = Lucas(n!).

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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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A261626 a(n) = Fibonacci(n!) - Fibonacci(n)!.

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A262776 a(n) = Fibonacci(n!) mod Fibonacci(n)!.

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A371136 Decimal expansion of Sum_{k>=1} 1/Fibonacci(k!).

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