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

1,2

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.

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

Example

1.410686134642447997690824711419115041323478...

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 *)

Prog

(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

Crossrefs

Cf. A001147, A112293, A286286.

Sequence in context: A011349 A303126 A292404 * A204169 A255331 A296794

Adjacent sequences: A060193 A060194 A060195 * A060197 A060198 A060199

Keyword

nonn,cons

Author

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

Extensions

More terms from Vladeta Jovovic, Mar 27 2001

Status

approved