A193027 Decimal expansion of the constant term of the reduction of e^(2x) by x^2->x+1.

7, 2, 3, 9, 8, 7, 7, 3, 3, 1, 3, 0, 0, 9, 0, 0, 1, 7, 7, 3, 2, 4, 9, 3, 2, 2, 2, 3, 3, 9, 6, 1, 6, 1, 0, 4, 2, 5, 9, 0, 1, 3, 3, 4, 6, 9, 1, 6, 7, 8, 8, 1, 3, 4, 6, 1, 0, 3, 0, 8, 8, 0, 9, 0, 1, 1, 9, 1, 0, 1, 2, 1, 1, 5, 7, 9, 3, 8, 8, 4, 9, 2, 0, 5, 8, 7, 9, 3, 1, 7, 7, 2, 9, 8, 3, 9, 6, 4, 3, 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 18 2022: (Start)

Equals 1 + Sum_{k>=1} 2^k*Fibonacci(k-1)/k!.

Equals e * (cosh(sqrt(5)) - sinh(sqrt(5))/sqrt(5)). (End)

Example

7.239877331300900177324932223396161042590133469167...

Mathematica

f[x_] := Exp[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, A193010, A192232, A193028.

Sequence in context: A019945 A335825 A336011 * A248288 A248287 A233929

Adjacent sequences: A193024 A193025 A193026 * A193028 A193029 A193030

Keyword

nonn,cons

Author

Clark Kimberling, Jul 14 2011

Status

approved