A322505 - OEIS

%I #8 May 24 2022 00:10:37

%S 0,1,1,0,4,5,0,6,4,9,0,11,7,3,11,10,2,2,5,16,11,3,7,18,16,19,11,12,21,

%T 19,22,5,31,21,25,30,20,6,5,21,17,41,36,14,28,13,45,16,0,33,1,2,41,1,

%U 28,43,9,15,16,28,22,19,22,13,34,61,38,40,56,44,69,25,42,44,34,73,71,42,17

%N Factorial expansion of 1/sqrt(2) = Sum_{n>=1} a(n)/n!.

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

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

%t With[{b = 1/Sqrt[2]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* _G. C. Greubel_, Dec 12 2018 *)

%o (PARI) default(realprecision, 250); b = 1/sqrt(2); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", "))

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

%o (Sage)

%o b=1/sqrt(2);

%o def a(n):

%o     if (n==1): return floor(b)

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

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

%Y Cf. A010503 (decimal expansion), A130130 (continued fraction).

%Y Cf. A009949 (sqrt(2)).

%K nonn

%O 1,5

%A _G. C. Greubel_, Dec 12 2018