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

%I #16 Sep 08 2022 08:45:05

%S 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,

%T 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,

%U 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

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

%H G. C. Greubel, <a href="/A068453/b068453.txt">Table of n, a(n) for n = 1..10000</a>

%t 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 *)

%o (PARI) vector(30,n,if(n>1,t=t%1*n,t=exp(.5))\1) \\ _M. F. Hasler_, Nov 25 2018

%o (PARI) 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

%o (Magma) 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

%o (Sage)

%o def A068453(n):

%o if (n==1): return floor(sqrt(e))

%o else: return expand(floor(factorial(n)*sqrt(e)) - n*floor(factorial(n-1)*sqrt(e)))

%o [A068453(n) for n in (1..80)] # _G. C. Greubel_, Nov 26 2018

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

%K easy,nonn

%O 1,4

%A _Benoit Cloitre_, Mar 10 2002

%E Name edited and keyword cons removed by _M. F. Hasler_, Nov 25 2018