A060196 Decimal expansion of 1 + 1/(1*3) + 1/(1*3*5) + 1/(1*3*5*7) + ...

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, 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, 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

Offset: 1

Author

Evan Michael Adams (evan(AT)tampabay.rr.com), Simon Plouffe, Mar 21 2001

Keywords

• nonn
• cons

Examples

			1.410686134642447997690824711419115041323478...
		

References

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

Links

• Harry J. Smith, Table of n, a(n) for n = 1..20000
• J.-P. Allouche and T. Baruchel, Variations on an error sum function for the convergents of some powers of e, arXiv preprint arXiv:1408.2206 [math.NT], 2014.

Crossrefs

Cf. A001147, A112293, A286286.

Programs

Mathematica

RealDigits[ Sqrt[E*Pi/2] * Erf[1/Sqrt[2]], 10, 107] // First
(* 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 *)

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

Formula

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

From Peter Bala, Feb 09 2024: (Start)

Generalized continued fraction expansion:

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

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)

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

Extensions

More terms from Vladeta Jovovic, Mar 27 2001