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A322505 Factorial expansion of 1/sqrt(2) = Sum_{n>=1} a(n)/n!.

%I A322505 #8 May 24 2022 00:10:37
%S A322505 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 A322505 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 A322505 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 A322505 Factorial expansion of 1/sqrt(2) = Sum_{n>=1} a(n)/n!.
%H A322505 <a href="https://oeis.org/index/Fa#facbase">Index entries for factorial base representation</a>
%e A322505 1/sqrt(2) = 0 + 1/2! + 1/3! + 0/4! + 4/5! + 5/6! + 0/7! + 6/8! + ...
%t A322505 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 A322505 (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 A322505 (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 A322505 (Sage)
%o A322505 b=1/sqrt(2);
%o A322505 def a(n):
%o A322505     if (n==1): return floor(b)
%o A322505     else: return expand(floor(factorial(n)*b) -n*floor(factorial(n-1)*b))
%o A322505 [a(n) for n in (1..80)]
%Y A322505 Cf. A010503 (decimal expansion), A130130 (continued fraction).
%Y A322505 Cf. A009949 (sqrt(2)).
%K A322505 nonn
%O A322505 1,5
%A A322505 _G. C. Greubel_, Dec 12 2018