cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 14 2011

Keywords

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

Examples

			1.783922996312878767846236916090170972510...

Crossrefs

Cf. A000045, A192232.

Programs

  • 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]

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)

A269574 Decimal expansion of Sum_{n>=1} (1-cos(Pi/n)).

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Mar 01 2016

Keywords

Comments

Value very close to A193081.

Examples

			4.87071896189479740325580288922801180768723798317416757630477557161789...

Crossrefs

Cf. A051762, A085365, A093721, A193081, A269611, A269720.

Programs

  • Maple
    evalf(Sum(1-cos(Pi/n), n=1..infinity), 120);
  • Mathematica
    RealDigits[NSum[1 - Cos[Pi/n], {n, 1, Infinity}, WorkingPrecision -> 120, NSumTerms -> 10000, PrecisionGoal -> 120, Method -> {"NIntegrate", "MaxRecursion" -> 100}]][[1]] (* Be aware that NSum[1 - Cos[Pi/n], {n, 1, Infinity}, WorkingPrecision -> 120] or N[Sum[1 - Cos[Pi/n], {n, 1, Infinity}], 120] give an incorrect numerical result (only 25 decimal places are correct!) *)
  • PARI
    default(realprecision,120); sumpos(n=1, 1-cos(Pi/n))

Formula

Equals 2 * Sum_{n>=1} (sin(Pi/(2*n)))^2.
Equals Sum_{k>=1} (-1)^(k+1) * Pi^(2*k) * Zeta(2*k) / (2*k)!, where Zeta is the Riemann zeta function.
Equals Sum_{k>=1} 2^(2*k-1) * Pi^(4*k) * B(2*k) / (2*k)!^2, where B(n) is the Bernoulli number A027641(n)/A027642(n).

A193082 Decimal expansion of the coefficient of x in the reduction of cosh(2x) by x^2->x+1.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 15 2011

Keywords

Comments

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

Examples

			4.86127014034021114230075809766492371...

Crossrefs

Cf. A000045, A193010, A192232, A193081.

Programs

  • Mathematica
    f[x_] := Cosh[2 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]

Formula

From Amiram Eldar, Jan 18 2022: (Start)
Equals Sum_{k>=1} 2^(2*k)*Fibonacci(2*k)/(2*k)!.
Equals 2*sinh(1)*sinh(sqrt(5))/sqrt(5). (End)
Showing 1-3 of 3 results.

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