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%I #35 Oct 02 2022 23:14:46
%S 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,
%T 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,
%U 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
%N Decimal expansion of 1/(e-2).
%C The value of the continued fraction 1+1/(2+2/(3+3/(4+4/(5+5/(6+6/(...)))))).
%H G. C. Greubel, <a href="/A194807/b194807.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F 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
%e 1.392211191177332814376552878479816528373978385315...
%t RealDigits[1/(E - 2), 10, 105][[1]] (* _T. D. Noe_, May 07 2012 *)
%t Fold[Function[#2 + #2/#1], 1, Reverse[Range[100]]] // N[#, 105]& // RealDigits // First (* _Jean-François Alcover_, Sep 19 2014 *)
%o (PARI)
%o default(realprecision,110);
%o 1/(exp(1)-2)
%o \\ _Joerg Arndt_, May 07 2012
%o (Magma) 1/(Exp(1) - 2); // _G. C. Greubel_, Apr 09 2018
%Y Cf. A073333 (1/(e-1)), A002627, A185108.
%K cons,easy,nonn
%O 1,2
%A _Martin Janecke_, May 06 2012
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