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.

Previous Showing 31-35 of 35 results.

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)

A193089 Decimal expansion of the constant term of the reduction of (sin(x))^2 by x^2->x+1.

Original entry on oeis.org

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
Offset: 0

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

			0.5187233386849291969930277770388656030...

Crossrefs

Cf. A000045, A001622, A193010, A192232, A193087, A193088.

Programs

  • 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]

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)

A193013 Decimal expansion of the constant term of the reduction of cos(x) by x^2->x+1.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 14 2011

Keywords

Comments

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

Examples

			0.5767021231946339181854043477561375348200573480...

Crossrefs

Cf. A000045, A193010, A192232, A193014.

Programs

  • Mathematica
    f[x_] := Cos[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, 300}], 100]
RealDigits[u0, 10]

Formula

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

A193033 Decimal expansion of the constant term of the reduction of 3^x by x^2->x+1.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 14 2011

Keywords

Comments

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

Examples

			2.00199585716593447105854181212612374067547375969106000...

Crossrefs

Cf. A000045, A001622, A193010, A192232, A193034.

Programs

  • Mathematica
    f[x_] := 3^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]

Formula

From Amiram Eldar, Jan 19 2022: (Start)
Equals 1 + Sum_{k>=1} log(3)^k*Fibonacci(k-1)/k!.
Equals (9^phi + 3*phi^2)/(sqrt(5)*phi*3^phi), where phi is the golden ratio (A001622). (End)

A193034 Decimal expansion of the coefficient of x in the reduction of 3^x by x^2->x+1.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 14 2011

Keywords

Comments

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

Examples

			2.4187370671895572561463726604300...

Crossrefs

Cf. A000045, A001622, A193010, A192232, A193033.

Programs

  • Mathematica
    f[x_] := 3^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 19 2022: (Start)
Equals Sum_{k>=0} log(3)^k*Fibonacci(k)/k!.
Equals (3^sqrt(5)-1)/(sqrt(5)*3^(phi-1)), where phi is the golden ratio (A001622). (End)
Previous Showing 31-35 of 35 results.

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