A193031 Decimal expansion of the constant term of the reduction of 2^x by x^2->x+1.

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

Offset

1,2

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(2)^k*Fibonacci(k-1)/k!.

Equals (1 + (3+4^phi)/sqrt(5))/(phi*2^phi), where phi is the golden ratio (A001622). (End)

Example

1.3198787240211562744599742126393139318590...

Mathematica

f[x_] := 2^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, A193032.

Sequence in context: A114195 A185948 A016601 * A078416 A369926 A223533

Adjacent sequences: A193028 A193029 A193030 * A193032 A193033 A193034

Keyword

nonn,cons

Author

Clark Kimberling, Jul 14 2011

Status

approved