A060196 - OEIS

%I #56 Dec 15 2024 15:43:41

%S 1,4,1,0,6,8,6,1,3,4,6,4,2,4,4,7,9,9,7,6,9,0,8,2,4,7,1,1,4,1,9,1,1,5,

%T 0,4,1,3,2,3,4,7,8,6,2,5,6,2,5,1,9,2,1,9,7,7,2,4,6,3,9,4,6,8,1,6,4,7,

%U 8,1,7,9,8,4,9,0,3,9,7,9,2,7,1,1,5,4,0,9,2,2,4,7,7,8,6,1,1,6,4,0,1,4,7,2,8,9,7

%N Decimal expansion of 1 + 1/(1*3) + 1/(1*3*5) + 1/(1*3*5*7) + ...

%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.2, p. 423.

%H Harry J. Smith, <a href="/A060196/b060196.txt">Table of n, a(n) for n = 1..20000</a>

%H J.-P. Allouche and T. Baruchel, <a href="http://arxiv.org/abs/1408.2206">Variations on an error sum function for the convergents of some powers of e</a>, arXiv preprint arXiv:1408.2206 [math.NT], 2014.

%F c = sqrt(e*Pi/2)*erf(1/sqrt(2)), or 2^(-1/2)*exp(1/2)*sqrt(Pi)*(1 - erfc(1/sqrt(2))). - _Michael Kleber_, Mar 21 2001

%F From _Peter Bala_, Feb 09 2024: (Start)

%F Generalized continued fraction expansion:

%F c = 1/(1 - 1/(4 - 3/(6 - 5/(8 - 7/(10 - 9/(12 - ... )))))). See A286286.

%F c/(1 + c) = Sum_{n >= 0} (2*n-1)!!/(A112293(n) * A112293(n+1)) = 1/(1*2) + 1/(2*7) + 3/(7*36) + 15/(36*253) + 105/(253*2278) + ... = 0.5851803411..., a rapidly converging series. (End)

%F Equals Sum_{n >= 0} ((n - 1)*(n + 1)!*2^(n + 1))/(2*n)!. - _Antonio Graciá Llorente_, Feb 13 2024

%e 1.410686134642447997690824711419115041323478...

%t RealDigits[ Sqrt[E*Pi/2] * Erf[1/Sqrt[2]], 10, 107] // First

%t (* or *) 1/Fold[Function[2*#2-1+(-1)^#2*#2/#1], 1, Reverse[Range[100]]] // N[#, 107]& // RealDigits // First (* _Jean-François Alcover_, Mar 07 2013, updated Sep 19 2014 *)

%o (PARI) { default(realprecision, 20080); x=2^(-1/2)*exp(1/2)*sqrt(Pi)*(1 - erfc(1/sqrt(2))); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b060196.txt", n, " ", d)); } \\ _Harry J. Smith_, Jul 02 2009

%Y Cf. A001147, A112293, A286286.

%K nonn,cons

%O 1,2

%A Evan Michael Adams (evan(AT)tampabay.rr.com), _Simon Plouffe_, Mar 21 2001

%E More terms from _Vladeta Jovovic_, Mar 27 2001