A322334 - OEIS

%I #17 May 24 2022 00:10:51

%S 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,

%T 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,

%U 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

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

%H G. C. Greubel, <a href="/A322334/b322334.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="https://oeis.org/index/Fa#facbase">Index entries for factorial base representation of noninteger constants</a>

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

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

%o (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)), ", "))

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

%o (Sage)

%o def a(n):

%o     if (n==1): return floor(log(3))

%o     else: return expand(floor(factorial(n)*log(3)) - n*floor(factorial(n-1)*log(3)))

%o [a(n) for n in (1..80)]

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

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

%K nonn

%O 1,4

%A _G. C. Greubel_, Dec 03 2018