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%I A193029 #11 Jan 18 2022 02:29:30
%S A193029 1,1,5,1,9,4,5,2,0,6,7,5,9,4,6,8,8,2,1,2,7,7,4,8,5,4,5,1,5,0,5,5,8,2,
%T A193029 7,4,3,2,1,2,3,8,5,8,9,0,4,1,1,3,1,5,1,1,6,6,5,2,0,0,0,1,1,8,0,1,6,4,
%U A193029 6,0,3,2,4,0,0,6,2,0,8,2,5,1,5,5,5,1,6,3,9,7,9,8,2,7,2,7,8,1,0,0
%N A193029 Decimal expansion of the constant term of the reduction of e^(x/2) by x^2->x+1.
%C A193029 Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010.
%F A193029 From _Amiram Eldar_, Jan 18 2022: (Start)
%F A193029 Equals 1 + Sum_{k>=1} Fibonacci(k-1)/(k!*2^k).
%F A193029 Equals exp(1/4)*(cosh(sqrt(5)/4) - sqrt(5)*sinh(sqrt(5)/4)/5). (End)
%e A193029 1.151945206759468821277485451505582743212385890...
%t A193029 f[x_] := Exp[x/2]; r[n_] := Fibonacci[n];
%t A193029 c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
%t A193029 u0 = N[Sum[c[n]*r[n - 1], {n, 0, 100}], 100]
%t A193029 RealDigits[u0, 10]
%Y A193029 Cf. A000045, A193010, A192232, A193030.
%K A193029 nonn,cons
%O A193029 1,3
%A A193029 _Clark Kimberling_, Jul 14 2011

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