A133658 Decimal expansion of Sum_{x=integer, -inf < x < inf} (1/sqrt(2*Pi))*exp(-x^2/2).

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

Offset: 1

Author

Martin Raab, Dec 28 2007

Keywords

• cons
• nonn

Comments

Standard normal distribution taken at all integers x from -infinity to +infinity.

Not only is this constant quite close to 1/tanh(pi^2) (difference is about 1.43*10^-17), but it is even closer if the second term of its continued fraction, 186895766.612113..., is reduced by 1/2 (the difference then decreases to about 10^-34).

The continued fraction begins: 1, 186895766, 1, 1, 1, 1, 2, 1, 2, 2, 1, 3, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 4, 1, 6, 1, 5, 8, 1, 1, 3, 1, 44, 3, 7, 31, 2, 5, 1, 1, 5, 1, 5, 5334, 1, ... - Robert G. Wilson v, Dec 30 2007

See A084304 for cont.frac.(1/tanh(pi^2)) = [1, 186895766, 8, 1, 11, 2, 3, ...] - M. F. Hasler, Oct 24 2009

Examples

			1.000000005350575982148479362482248...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000

Programs

Mathematica

RealDigits[(1 + 2*Sum[ Exp[ -x^2/2], {x, 1, 24, 1}])/Sqrt[2 Pi], 10, 2^7][[1]] (* Robert G. Wilson v, Dec 30 2007 *)

PARI

default(realprecision,100); sqrt(2/Pi)*(suminf(k=1,exp(-k^2/2))+.5)
vecextract(eval(Vec(Str( % ))),"^2") \\ M. F. Hasler, Oct 24 2009

Extensions

More terms from Robert G. Wilson v, Dec 30 2007