A193075 -id:A193075 - cp's OEIS frontend

A193010 Decimal expansion of the constant term of the reduction of e^x by x^2->x+1.

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

Offset: 1

Clark Kimberling, Jul 14 2011

Suppose that q and s are polynomials and degree(q)>degree(s).  The reduction of a polynomial p by q->s is introduced at A192232.  If p is replaced by a function f having power series
c(0) + c(1)*x + c(2)*x^2 + ... ,
then the reduction, R(f), of f by q->s is here introduced as the limit, if it exists, of the reduction of p(n,x) by q->s, where p(n,x) is the n-th partial sum of f(x):
R(f(x)) = c(0)*R(1) + c(1)*R(x) + c(2)*R(x^2) + ...  If q(x)=x^2 and s(x)=x+1, then
R(f(x)) = c(0) + c(1)*x + c(2)*(x+1) + c(3)*(2x+1) + c(4)(3x+2) + ..., so that
R(f(x)) = Sum_{n>=0} c(n)*(F(n)*x+F(n-1)), where F=A000045 (Fibonacci sequence), so that
R(f(x)) = u0 + x*u1 where u0 = Sum_{n>=0} c(n)*F(n-1), u1 = Sum_{n>=0} c(n)*F(n); the numbers u0 and u1 are given by A193010 and A098689.
Following is a list of reductions by x^2->x+1 of selected functions.  Each sequence A-number refers to the constant represented by the sequence.  Adjustments for offsets are needed in some cases.
e^x......... A193010 + x*A098689
e^(-x)...... A193026 + x*A099935
e^(2x)...... A193027 + x*A193028
e^(x/2)..... A193029 + x*A193030
sin x....... A193011 + x*A193012
cos x....... A193013 + x*A193014
sinh x...... A193015 + x*A193016
cosh x...... A193017 + x*A193025
2^x......... A193031 + x*A193032
2^(-x)...... A193009 + x*A193035
3^x......... A193083 + x*A193084
t^x......... A193075 + x*A193076, t=(1+sqrt(5))/2
t^(-x)...... A193077 + x*A193078, t=(1+sqrt(5))/2
sinh(2x).... A193079 + x*A193080
cosh(2x).... A193081 + x*A193082
(e^x)cos x.. A193083 + x*A193084
(e^x)sin x.. A193085 + x*A193086
(cos x)^2... A193087 + x*A193088
(sin x)^2... A193089 + x*A193088

			1.783922996312878767846236916090170972510...

Cf. A000045, A192232.

f[x_] := Exp[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, 200}], 100]
RealDigits[u0, 10]

From Amiram Eldar, Jan 18 2022: (Start)
Equals 1 + Sum_{k>=1} Fibonacci(k-1)/k!.
Equals (sqrt(5)-1) * (2*sqrt(5)*exp(sqrt(5)) + 3*sqrt(5) + 5) / (20 * exp((sqrt(5)-1)/2)). (End)

A193076 Decimal expansion of the coefficient of x in the reduction of phi^x by x^2->x+1, where phi=(1+sqrt(5))/2 is the golden ratio (A001622).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Jul 15 2011

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

			0.6420710988036375722663484493183969433220...

Cf. A000045, A001622, A193010, A192232, A193075.

t = GoldenRatio
f[x_] := t^(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]

From Amiram Eldar, Jan 19 2022: (Start)
Equals Sum_{k>=0} log(phi)^k*Fibonacci(k)/k!.
Equals (phi^phi - phi^(1-phi))/sqrt(5). (End)

a(99)-a(100) from Georg Fischer, Feb 08 2025

cp's OEIS Frontend

A193010 Decimal expansion of the constant term of the reduction of e^x by x^2->x+1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula