A099554 Decimal expansion of the constant x that satisfies x = exp(1/sqrt(x)).

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

Offset: 1

Author

Paul D. Hanna, Oct 22 2004

Keywords

• cons
• nonn

Comments

This constant arises from the series: S(n) = Sum_{k=0..2n} (n-[k/2])^k/k!. The asymptotic behavior of this series is given by: S(n) ~ c*x^n where c = (x+sqrt(x))/(1+2*sqrt(x)) = 0.8957126... and x = 2.0207473586... satisfies x = exp(1/sqrt(x)).

Examples

			x=2.02074735861185766811269528724732366499433113141625298973171160826928577...
To demonstrate how this constant describes the asymptotics of the sum:
S(n) = Sum_{k=0..2n} (n-[k/2])^k/k! ~ c*x^n
evaluate the sum at n=5:
S(5) = 1+ 5+ 4^2/2!+ 4^3/3!+ 3^4/4!+ 3^5/5!+ 2^6/6!+ 2^7/7!+ 1/8!+ 1/9!
= 782291/25920 = 30.1809799... = (0.89572199...)*x^5
and evaluate the sum at n=6:
S(6) = 1+ 6+ 5^2/2!+ 5^3/3!+ 4^4/4!+ 4^5/5!+ 3^6/6!+ 3^7/7!+ 2^8/8!+ 2^9/9!+ 1/10!+ 1/11!
= 608606683/9979200 = 60.9875223... = (0.89571298...)*x^6.
		

Crossrefs

Cf. A099555, A202356.

Programs

Mathematica

RealDigits[x/.FindRoot[x==Exp[1/Sqrt[x]],{x,2},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Jan 06 2013 *)
RealDigits[ 1/(4*ProductLog[1/2]^2), 10, 105] // First (* Jean-François Alcover, Feb 15 2013 *)

PARI

solve(x=2,2.1,x-exp(1/sqrt(x)))

Formula

Equals 1/(4*A202356^2). - Vaclav Kotesovec, Oct 06 2020