A067881 Factorial expansion of sqrt(3) = Sum_{n>=1} a(n)/n!.

1, 1, 1, 1, 2, 5, 0, 4, 2, 5, 10, 8, 1, 5, 6, 8, 5, 13, 18, 0, 7, 20, 9, 6, 14, 2, 7, 7, 18, 11, 0, 12, 20, 10, 31, 28, 27, 34, 29, 18, 13, 8, 28, 14, 9, 12, 39, 5, 15, 8, 5, 0, 7, 21, 54, 13, 16, 20, 24, 18, 12, 14, 6, 53, 21, 42, 47, 14, 46, 14, 42, 71, 41, 63, 24, 28, 32, 61, 35

Offset: 1

Author

Benoit Cloitre, Mar 10 2002

Keywords

• easy
• nonn

Examples

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

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Index entries for factorial base representation

Crossrefs

Cf. A002194 (decimal expansion), A040001 (continued fraction).

Cf. A009949 (sqrt(2)), A068446 (sqrt(5)), A320839 (sqrt(7)).

Programs

Magma

SetDefaultRealField(RealField(250));  [Floor(Sqrt(3))] cat [Floor(Factorial(n)*Sqrt(3)) - n*Floor(Factorial((n-1))*Sqrt(3)) : n in [2..80]]; // G. C. Greubel, Dec 10 2018

Maple

Digits:=200: a:=n->`if`(n=1,floor(sqrt(3)),floor(factorial(n)*sqrt(3))-n*floor(factorial(n-1)*sqrt(3))): seq(a(n),n=1..90); # Muniru A Asiru, Dec 11 2018

Mathematica

With[{b = Sqrt[3]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Dec 10 2018 *)

PARI

default(realprecision, 250); {b = sqrt(3); a(n) = if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b))};
for(n=1, 80, print1(a(n), ", ")) \\ G. C. Greubel, Dec 10 2018

PARI

apply( A067881(n)=if(n>1,sqrt(precision(3., n*log(n/2.5)\2.3+2))*(n-1)!%1*n\1,1), [1..79]) \\ M. F. Hasler, Dec 14 2018

Sage

b=sqrt(3);
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)] # G. C. Greubel, Dec 10 2018

Formula

a(1) = 1; for n > 1, a(n) = floor(n!*sqrt(3)) - n*floor((n-1)!*sqrt(3)).