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

%I A322506 #8 May 24 2022 00:10:27
%S A322506 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,
%T A322506 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,
%U A322506 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
%N A322506 Factorial expansion of 1/exp(2) = Sum_{n>=1} a(n)/n!.
%H A322506 <a href="https://oeis.org/index/Fa#facbase">Index entries for factorial base representation</a>
%e A322506 1/exp(2) = 0 + 0/2! + 0/3! + 3/4! + 1/5! + 1/6! + 3/7! + 0/8! + 6/9! +...
%t A322506 With[{b = 1/E^2}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]]
%o A322506 (PARI) default(realprecision, 250); b = exp(-2); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", "))
%o A322506 (Magma) SetDefaultRealField(RealField(250));  [Floor(Exp(-2))] cat [Floor(Factorial(n)*Exp(-2)) - n*Floor(Factorial((n-1))*Exp(-2)) : n in [2..80]];
%o A322506 (Sage)
%o A322506 b=exp(-2);
%o A322506 def a(n):
%o A322506     if (n==1): return floor(b)
%o A322506     else: return expand(floor(factorial(n)*b) -n*floor(factorial(n-1)*b))
%o A322506 [a(n) for n in (1..80)]
%Y A322506 Cf. A092553 (decimal expansion), 0 U A001204 (continued fraction).
%Y A322506 Cf. A054977 (e), A067840 (e^2), A068453 (sqrt(e)), A237420 (1/e).
%K A322506 nonn
%O A322506 1,4
%A A322506 _G. C. Greubel_, Dec 12 2018