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%I #22 Sep 08 2022 08:46:17
%S 2,0,2,8,1,1,4,9,8,1,6,4,7,4,7,2,4,5,1,1,0,8,1,2,6,1,1,2,7,4,6,3,5,1,
%T 1,7,5,1,7,4,3,2,5,0,9,2,5,4,2,6,1,3,5,2,0,6,1,7,7,7,5,9,7,2,1,2,2,2,
%U 1,5,3,9,5,0,4,8,7,1,6,5,5,9,4,2,5,9,6
%N Decimal expansion of exp(sqrt(2)/2).
%C 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.
%C We find that C2 = lim_{n->infinity} P(n) = exp(sqrt(2)/2).
%C The structure of the n!*P(n) formulas leads to the multinomial coefficients A036039.
%H G. C. Greubel, <a href="/A274541/b274541.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)/2).
%F 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.
%e c = 2.02811498164747245110812611274635117517432509254...
%p Digits := 140: evalf(exp(sqrt(2)/2)); # 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=2 then (sqrt(2)+1) else 1 fi: end:
%p Digits := 140: evalf(P(250)); # End program 2.
%t First@ RealDigits@ N[Exp[Sqrt[2]/2], 83] (* _Michael De Vlieger_, Jun 27 2016 *)
%o (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
%o (Magma) SetDefaultRealField(RealField(100)); Exp[Sqrt[2]/2]; // _G. C. Greubel_, Aug 19 2018
%Y Cf. A274540, A274542, A014176, A010503, A036039.
%K cons,nonn
%O 1,1
%A _Johannes W. Meijer_, Jun 27 2016
%E More digits from _Jon E. Schoenfield_, Mar 15 2018
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