A322334 Factorial expansion of log(3) = Sum_{n>=1} a(n)/n!.

1, 0, 0, 2, 1, 5, 0, 0, 0, 4, 3, 0, 0, 9, 6, 1, 11, 4, 13, 8, 9, 0, 16, 2, 14, 24, 9, 22, 5, 26, 4, 2, 31, 15, 17, 15, 8, 31, 18, 17, 20, 36, 20, 3, 41, 12, 7, 44, 44, 2, 38, 20, 44, 47, 3, 44, 19, 40, 9, 14, 1, 24, 15, 46, 0, 60, 37, 67, 63, 24, 64, 51, 30, 31, 59, 18, 68, 63, 22, 16, 45, 29, 43, 24, 13, 26, 77, 30, 37, 41, 3, 29, 25, 88, 12, 93, 56, 60, 60, 13

Offset: 1

Author

G. C. Greubel, Dec 03 2018

Keywords

• nonn

Examples

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

Links

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

Crossrefs

Cf. A002391 (decimal expansion), A016731 (continued fraction).

Cf. A067882 (log(2)), A322333 (log(5)), A068460 (log(7)), A068461 (log(11)).

Programs

Magma

SetDefaultRealField(RealField(250));  [Floor(Log(3))] cat [Floor(Factorial(n)*Log(3)) - n*Floor(Factorial((n-1))*Log(3)) : n in [2..80]];

Mathematica

With[{b = Log[3]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]]

PARI

default(realprecision, 250); b = log(3); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", "))

Sage

def a(n):
    if (n==1): return floor(log(3))
    else: return expand(floor(factorial(n)*log(3)) - n*floor(factorial(n-1)*log(3)))
[a(n) for n in (1..80)]