A193030 - OEIS

%I #11 Jan 18 2022 02:29:51

%S 6,7,5,9,7,7,2,4,5,8,7,2,0,5,1,0,7,7,6,6,2,2,5,9,6,3,7,4,1,7,5,6,3,0,

%T 7,0,4,1,7,1,2,0,8,6,0,5,3,2,6,1,6,1,7,4,0,0,2,1,3,8,5,4,2,3,1,3,6,0,

%U 2,9,1,8,9,5,2,8,7,7,5,3,2,1,1,4,2,1,8,5,4,2,6,8,5,0,3,7,6,6,0,9

%N Decimal expansion of the coefficient of x in the reduction of e^(x/2) by x^2->x+1.

%C Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010.

%F From _Amiram Eldar_, Jan 18 2022: (Start)

%F Equals Sum_{k>=1} Fibonacci(k)/(k!*2^k).

%F Equals 2*exp(1/4)*sinh(sqrt(5)/4)/sqrt(5). (End)

%e 0.675977245872051077662259637417563070...

%t f[x_] := Exp[x/2]; r[n_] := Fibonacci[n];

%t c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]

%t u1 = N[Sum[c[n]*r[n], {n, 0, 100}], 100]

%t RealDigits[u1, 10]

%Y Cf. A000045, A193010, A192232, A193029.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Jul 14 2011