OFFSET
1,1
COMMENTS
Define P(n) = (1/n)*Sum_{k=0..n-1} x(n-k)*P(k), n >= 1 and P(0) = 1 with x(2) = (sqrt(2) + 1) and x(n) = 1 for all other n.
We find that C2 = lim_{n->infinity} P(n) = exp(sqrt(2)/2).
The structure of the n!*P(n) formulas leads to the multinomial coefficients A036039.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..10000
The Dev Team and Simon Plouffe, The Inverse Symbolic Calculator (ISC).
FORMULA
c = exp(sqrt(2)/2).
c = lim_{n->infinity} P(n), with P(n) = (1/n)*Sum_{k=0..n-1} x(n-k)*P(k), for n >= 1, and P(0) = 1, with x(2) = (1 + sqrt(2)), the silver mean A014176, and x(n) = 1 for all other n.
EXAMPLE
c = 2.02811498164747245110812611274635117517432509254...
MAPLE
Digits := 140: evalf(exp(sqrt(2)/2)); # End program 1.
P := proc(n) : if n=0 then 1 else P(n) := expand((1/n)*(add(x(n-k)*P(k), k=0..n-1))) fi; end: x := proc(n): if n=2 then (sqrt(2)+1) else 1 fi: end:
Digits := 140: evalf(P(250)); # End program 2.
MATHEMATICA
First@ RealDigits@ N[Exp[Sqrt[2]/2], 83] (* Michael De Vlieger, Jun 27 2016 *)
PROG
(PARI) my(x=exp(sqrt(2)/2)); for(k=1, 100, my(d=floor(x)); x=(x-d)*10; print1(d, ", ")) \\ Felix Fröhlich, Jun 27 2016
(Magma) SetDefaultRealField(RealField(100)); Exp[Sqrt[2]/2]; // G. C. Greubel, Aug 19 2018
CROSSREFS
KEYWORD
AUTHOR
Johannes W. Meijer, Jun 27 2016
EXTENSIONS
More digits from Jon E. Schoenfield, Mar 15 2018
STATUS
approved