A193032 - OEIS

%I #13 Jan 19 2022 05:37:33

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

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

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

%N Decimal expansion of the coefficient of x in the reduction of 2^x 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 19 2022: (Start)

%F Equals Sum_{k>=0} log(2)^k*Fibonacci(k)/k!.

%F Equals (2^sqrt(5)-1)/(sqrt(5)*2^(phi-1)), where phi is the golden ratio (A001622). (End)

%e 1.081365283916960767554821105448442602497...

%t f[x_] := 2^x; 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, A001622, A193010, A192232, A193031.

%K nonn,cons

%O 1,3

%A _Clark Kimberling_, Jul 14 2011