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%I #13 Sep 08 2022 08:46:17
%S 1,6,0,2,2,4,2,9,9,7,2,0,3,5,6,0,1,5,0,9,9,5,1,7,7,7,7,2,2,2,8,6,7,8,
%T 7,5,8,5,1,2,9,6,1,6,8,2,9,5,4,5,4,7,1,8,7,4,2,6,8,2,2,4,0,5,3,0,9,1,
%U 0,0,1,6,9,9,4,9,0,4,1,9,1,9,5,8,2
%N Decimal expansion of exp(sqrt(2)/3).
%C Define P(n) = (1/n)*(sum(x(n-k)*P(k), k=0..n-1)), n >= 1 and P(0) =1 with x(3) = (1 + sqrt(2)) and x(n) = 1 for all other n. We find that C2 = limit(P(n), n -> infinity) = exp(sqrt(2)/3).
%C The structure of the n!*P(n) formulas leads to the multinomial coefficients A036039.
%H G. C. Greubel, <a href="/A274542/b274542.txt">Table of n, a(n) for n = 1..10000</a>
%H The Dev Team and Simon Plouffe, <a href="http://isc.carma.newcastle.edu.au/">The Inverse Symbolic Calculator (ISC)</a>.
%F c = exp(sqrt(2)/3)
%F c = limit(P(n), n -> infinity) with P(n) = (1/n)*(sum(x(n-k)*P(k), k=0..n-1)) for n >= 1, and P(0) =1, with x(3) = (1 + sqrt(2)), the silver mean A014176, and x(n) = 1 for all other n.
%e c = 1.6022429972035601509951777722286787585129616829545471874……
%p Digits := 85: evalf(exp(sqrt(2)/3)); # End program 1.
%p 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=3 then (sqrt(2)+1) else 1 fi: end: Digits := 56; evalf(P(120)); # End program 2.
%t First@ RealDigits@ N[Exp[Sqrt[2]/3], 85] (* _Michael De Vlieger_, Jun 27 2016 *)
%o (PARI) my(x=exp(sqrt(2)/3)); for(k=1, 100, my(d=floor(x)); x=(x-d)*10; print1(d, ", ")) \\ _Felix Fröhlich_, Jun 27 2016
%o (Magma) SetDefaultRealField(RealField(100)); Exp[Sqrt[2]/3]; // _G. C. Greubel_, Aug 19 2018
%Y Cf. A274540, A274541, A014176, A131594, A036039.
%K cons,nonn
%O 1,2
%A _Johannes W. Meijer_, Jun 27 2016
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