A193034 Decimal expansion of the coefficient of x in the reduction of 3^x by x^2->x+1.

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

Offset

1,1

Comments

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

Links

Table of n, a(n) for n=1..100.

Formula

From Amiram Eldar, Jan 19 2022: (Start)

Equals Sum_{k>=0} log(3)^k*Fibonacci(k)/k!.

Equals (3^sqrt(5)-1)/(sqrt(5)*3^(phi-1)), where phi is the golden ratio (A001622). (End)

Example

2.4187370671895572561463726604300...

Mathematica

f[x_] := 3^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]

Crossrefs

Cf. A000045, A001622, A193010, A192232, A193033.

Sequence in context: A345881 A091894 A127151 * A254179 A328641 A208741

Adjacent sequences: A193031 A193032 A193033 * A193035 A193036 A193037

Keyword

nonn,cons

Author

Clark Kimberling, Jul 14 2011

Status

approved