A193032 Decimal expansion of the coefficient of x in the reduction of 2^x by x^2->x+1.

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, 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, 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

Offset: 1

Author

Clark Kimberling, Jul 14 2011

Keywords

• nonn
• cons

Comments

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

Examples

			1.081365283916960767554821105448442602497...
		

Crossrefs

Cf. A000045, A001622, A193010, A192232, A193031.

Programs

Mathematica

f[x_] := 2^x; r[n_] := Fibonacci[n];
c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
u1 = N[Sum[c[n]*r[n], {n, 0, 100}], 100]
RealDigits[u1, 10]

Formula

From Amiram Eldar, Jan 19 2022: (Start)

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

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