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

1,2

Comments

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

Links

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

Formula

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)

Example

1.783922996312878767846236916090170972510...

Mathematica

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]

Crossrefs

Cf. A000045, A192232.

Sequence in context: A133613 A296140 A194622 * A079082 A091683 A359009

Adjacent sequences: A193007 A193008 A193009 * A193011 A193012 A193013

Keyword

nonn,cons

Author

Clark Kimberling, Jul 14 2011

Status

approved