A067840 -id:A067840 - cp's OEIS frontend

A068453 Factorial expansion of sqrt(e) = Sum_{n>=1} a(n)/n!.

1, 1, 0, 3, 2, 5, 0, 4, 3, 9, 8, 2, 8, 0, 10, 15, 2, 10, 8, 19, 12, 4, 18, 23, 8, 4, 21, 15, 17, 1, 11, 19, 7, 25, 15, 3, 20, 5, 24, 25, 35, 9, 12, 25, 26, 22, 23, 11, 43, 46, 6, 0, 25, 27, 30, 6, 14, 20, 33, 5, 30, 23, 42, 4, 11, 19, 55, 63, 43, 12, 52, 51, 22, 29, 11, 8, 19, 35, 25

Offset: 1

Benoit Cloitre, Mar 10 2002

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A067840 (e^2), A075874 (Pi).

SetDefaultRealField(RealField(250)); [Floor(Sqrt(Exp(1)))] cat [Floor(Factorial(n)*Sqrt(Exp(1))) - n*Floor(Factorial((n-1))* Sqrt(Exp(1))) : n in [2..80]]; // G. C. Greubel, Nov 26 2018

With[{b = Sqrt[E]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Nov 26 2018 *)

vector(30,n,if(n>1,t=t%1*n,t=exp(.5))\1) \\ M. F. Hasler, Nov 25 2018

default(realprecision, 250); b = sqrt(exp(1)); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", ")) \\ G. C. Greubel, Nov 26 2018

def A068453(n):
    if (n==1): return floor(sqrt(e))
    else: return expand(floor(factorial(n)*sqrt(e)) - n*floor(factorial(n-1)*sqrt(e)))
[A068453(n) for n in (1..80)] # G. C. Greubel, Nov 26 2018

Name edited and keyword cons removed by M. F. Hasler, Nov 25 2018

A322506 Factorial expansion of 1/exp(2) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

0, 0, 0, 3, 1, 1, 3, 0, 6, 4, 7, 5, 2, 9, 9, 8, 10, 8, 9, 1, 13, 18, 1, 2, 8, 15, 26, 10, 22, 1, 18, 9, 20, 10, 2, 6, 13, 19, 16, 38, 38, 3, 32, 5, 39, 24, 7, 27, 14, 41, 20, 39, 32, 7, 20, 35, 44, 50, 24, 34, 51, 14, 39, 47, 49, 15, 61, 54, 60, 52, 34, 60, 32, 72, 48, 12, 67, 52, 22, 48

Offset: 1

G. C. Greubel, Dec 12 2018

nonn

			1/exp(2) = 0 + 0/2! + 0/3! + 3/4! + 1/5! + 1/6! + 3/7! + 0/8! + 6/9! +...

Index entries for factorial base representation

Cf. A092553 (decimal expansion), 0 U A001204 (continued fraction).

Cf. A054977 (e), A067840 (e^2), A068453 (sqrt(e)), A237420 (1/e).

SetDefaultRealField(RealField(250));  [Floor(Exp(-2))] cat [Floor(Factorial(n)*Exp(-2)) - n*Floor(Factorial((n-1))*Exp(-2)) : n in [2..80]];

With[{b = 1/E^2}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]]

default(realprecision, 250); b = exp(-2); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", "))

b=exp(-2);
def a(n):
    if (n==1): return floor(b)
    else: return expand(floor(factorial(n)*b) -n*floor(factorial(n-1)*b))
[a(n) for n in (1..80)]

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A068453 Factorial expansion of sqrt(e) = Sum_{n>=1} a(n)/n!.

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A322506 Factorial expansion of 1/exp(2) = Sum_{n>=1} a(n)/n!.

Views

Author

Keywords

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Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage