A099554 - OEIS

%I #10 Oct 06 2020 02:39:37

%S 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,

%T 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,

%U 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

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

%C 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)).

%F Equals 1/(4*A202356^2). - _Vaclav Kotesovec_, Oct 06 2020

%e x=2.02074735861185766811269528724732366499433113141625298973171160826928577...

%e To demonstrate how this constant describes the asymptotics of the sum:

%e S(n) = Sum_{k=0..2n} (n-[k/2])^k/k! ~ c*x^n

%e evaluate the sum at n=5:

%e 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!

%e = 782291/25920 = 30.1809799... = (0.89572199...)*x^5

%e and evaluate the sum at n=6:

%e 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!

%e = 608606683/9979200 = 60.9875223... = (0.89571298...)*x^6.

%t RealDigits[x/.FindRoot[x==Exp[1/Sqrt[x]],{x,2},WorkingPrecision->120]][[1]] (* _Harvey P. Dale_, Jan 06 2013 *)

%t RealDigits[ 1/(4*ProductLog[1/2]^2), 10, 105] // First (* _Jean-François Alcover_, Feb 15 2013 *)

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

%Y Cf. A099555, A202356.

%K cons,nonn

%O 1,1

%A _Paul D. Hanna_, Oct 22 2004