A193033 Decimal expansion of the constant term of the reduction of 3^x by x^2->x+1.

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

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 1 + Sum_{k>=1} log(3)^k*Fibonacci(k-1)/k!.

Equals (9^phi + 3*phi^2)/(sqrt(5)*phi*3^phi), where phi is the golden ratio (A001622). (End)

Example

2.00199585716593447105854181212612374067547375969106000...

Mathematica

f[x_] := 3^x; r[n_] := Fibonacci[n];
c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
u0 = N[Sum[c[n]*r[n - 1], {n, 0, 100}], 100]
RealDigits[u0, 10]

Crossrefs

Cf. A000045, A001622, A193010, A192232, A193034.

Sequence in context: A332409 A367000 A289084 * A318253 A249772 A193531

Adjacent sequences: A193030 A193031 A193032 * A193034 A193035 A193036

Keyword

nonn,cons

Author

Clark Kimberling, Jul 14 2011

Status

approved