A194807 - OEIS
1, 3, 9, 2, 2, 1, 1, 1, 9, 1, 1, 7, 7, 3, 3, 2, 8, 1, 4, 3, 7, 6, 5, 5, 2, 8, 7, 8, 4, 7, 9, 8, 1, 6, 5, 2, 8, 3, 7, 3, 9, 7, 8, 3, 8, 5, 3, 1, 5, 2, 8, 7, 1, 2, 3, 5, 9, 1, 3, 2, 4, 5, 6, 7, 0, 8, 3, 2, 7, 9, 5, 7, 0, 4, 6, 1, 6, 1, 0, 9, 2, 6, 6, 9, 1, 7, 1, 0, 5, 8, 7, 2, 6, 7, 6, 1, 2, 9, 9, 8, 8, 8, 8, 5, 6
COMMENTS
The value of the continued fraction 1+1/(2+2/(3+3/(4+4/(5+5/(6+6/(...)))))).
FORMULA
Define s(n) = Sum_{k = 2..n} 1/k! for n >= 2. Then 1/(e - 2) = 2! - Sum_ {n >= 2} 1/( (n+1)!*s(n)*s(n+1) ) is a rapidly converging series of rationals. Cf. A073333. Equivalently, 1/(e - 2) = 2! - 2!/(1*4) - 3!/(4*17) - 4!/(17*86) - ..., where [1, 4, 17, 86, ... ] is A056542. Cf. A002627 and A185108. - Peter Bala, Oct 09 2013
EXAMPLE
1.392211191177332814376552878479816528373978385315...
MATHEMATICA
RealDigits[1/(E - 2), 10, 105][[1]] (* T. D. Noe, May 07 2012 *)
Fold[Function[#2 + #2/#1], 1, Reverse[Range[100]]] // N[#, 105]& // RealDigits // First (* Jean-François Alcover, Sep 19 2014 *)
PROG
(PARI)
default(realprecision, 110);
1/(exp(1)-2)
(Magma) 1/(Exp(1) - 2); // G. C. Greubel, Apr 09 2018