A193088 -id:A193088

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A193010

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

+10
36

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

(list; constant; graph; refs; listen; history; text; internal format)

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.

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Jul 14 2011

STATUS

approved

A193087

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

+10
3

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

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=0..99.

FORMULA

From Amiram Eldar, Jan 19 2022: (Start)

Equals 1 - A193089.

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

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

EXAMPLE

0.48127666131507080300697222296113439695...

MATHEMATICA

f[x_] := Cos[x]^2; 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, A193088, A193089.

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Jul 15 2011

STATUS

approved

A193089

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

+10
2

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

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=0..99.

FORMULA

From Amiram Eldar, Jan 19 2022: (Start)

Equals 1 - A193087.

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

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

EXAMPLE

0.5187233386849291969930277770388656030...

MATHEMATICA

f[x_] := Sin[x]^2; 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, A193087, A193088.

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Jul 15 2011

STATUS

approved

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