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 11-20 of 35 results. Next

A193011 Decimal expansion of the constant term of the reduction of sin(x) by x^2->x+1.

Original entry on oeis.org

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

			-0.143197452910080232641503647724641084990...

Crossrefs

Cf. A000045, A193010, A192232, A193012.

Programs

  • Mathematica
    f[x_] := Sin[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 Sum_{k>=1} (-1)^k*Fibonacci(2*k)/(2*k+1)!.
Equals cos(sqrt(5)/2)*sin(1/2) - cos(1/2)*sin(sqrt(5)/2)/sqrt(5). (End)

A193012 Decimal expansion of the coefficient of x in the reduction of sin(x) by x^2->x+1.

Original entry on oeis.org

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

			0.705845470457234393737808753502039387530431888887345001...

Crossrefs

Cf. A000045, A193010, A193011, A192232.

Programs

  • Mathematica
    f[x_] := Sin[x]; r[n_] := Fibonacci[n];
c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
u1 = N[Sum[c[n]*r[n], {n, 0, 200}], 100]
RealDigits[u1, 10]

Formula

From Amiram Eldar, Jan 18 2022: (Start)
Equals Sum_{k>=0} (-1)^k*Fibonacci(2*k+1)/(2*k+1)!.
Equals 2*cos(1/2)*sin(sqrt(5)/2)/sqrt(5). (End)

A193015 Decimal expansion of the constant term of the reduction of sinh(x) by x^2->x+1.

Original entry on oeis.org

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

			0.1933132399064173048764437960905880...

Crossrefs

Cf. A000045, A001622, A193010, A192232, A193016.

Programs

  • Mathematica
    f[x_] := Sinh[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 Sum_{k>=1} Fibonacci(2*k)/(2*k+1)!.
Equals (2 * sinh(phi) - (3+sqrt(5))*sinh(1/phi))/(5+sqrt(5)), where phi is the golden ratio (A001622). (End)

A193016 Decimal expansion of the coefficient of x in the reduction of sinh(x) by x^2->x+1.

Original entry on oeis.org

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

			1.3776753274909465462115651210703917736958351560...

Crossrefs

Cf. A000045, A193010, A192232, A193015.

Programs

  • Mathematica
    f[x_] := Sinh[x]; r[n_] := Fibonacci[n];
c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
u1 = N[Sum[c[n]*r[n], {n, 0, 300}], 100]
RealDigits[u1, 10]

Formula

From Amiram Eldar, Jan 18 2022: (Start)
Equals Sum_{k>=0} Fibonacci(2*k+1)/(2*k+1)!.
Equals (e+1)*sinh(sqrt(5)/2)/sqrt(5*e). (End)

A193017 Decimal expansion of the constant term of the reduction of cosh(x) by x^2->x+1.

Original entry on oeis.org

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

			1.5906097564064614629697931199995829165183536750...

Crossrefs

Cf. A000045, A001622, A193010, A192232, A193025.

Programs

  • Mathematica
    f[x_] := Cosh[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} Fibonacci(2*k-1)/(2*k)!.
Equals (2 * cosh(phi) + (3+sqrt(5))*cosh(1/phi))/(5+sqrt(5)), where phi is the golden ratio (A001622). (End)

A193025 Decimal expansion of the coefficient of x in the reduction of cosh(x) by x^2->x+1.

Original entry on oeis.org

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

			0.6366474059673691903697811343993673619632...

Crossrefs

Cf. A000045, A193010, A192232, A193017.

Programs

  • Mathematica
    f[x_] := Cosh[x]; r[n_] := Fibonacci[n];
c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
u1 = N[Sum[c[n]*r[n], {n, 0, 300}], 100]
RealDigits[u1, 10]

Formula

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

A193026 Decimal expansion of the constant term of the reduction of e^(-x) by x^2->x+1.

Original entry on oeis.org

1, 3, 9, 7, 2, 9, 6, 5, 1, 6, 5, 0, 0, 0, 4, 4, 1, 5, 8, 0, 9, 3, 3, 4, 9, 3, 2, 3, 9, 0, 8, 9, 9, 4, 8, 6, 0, 5, 2, 6, 4, 0, 8, 7, 4, 3, 7, 2, 3, 7, 0, 9, 2, 3, 3, 5, 6, 4, 0, 8, 2, 8, 9, 0, 2, 5, 9, 3, 6, 7, 5, 9, 2, 4, 7, 1, 6, 5, 8, 7, 6, 7, 5, 3, 6, 4, 1, 3, 7, 5, 5, 7, 8, 3, 4, 4, 0, 2, 4, 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. The coefficient of x in this reduction is the constant at A099935.

Examples

			1.39729651650004415809334932390899486052640...

Crossrefs

Cf. A000045, A193010, A192232, A099935.

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, 100}], 100]
RealDigits[u0, 10]

Formula

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

A193027 Decimal expansion of the constant term of the reduction of e^(2x) by x^2->x+1.

Original entry on oeis.org

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

			7.239877331300900177324932223396161042590133469167...

Crossrefs

Cf. A000045, A193010, A192232, A193028.

Programs

  • Mathematica
    f[x_] := Exp[2 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 18 2022: (Start)
Equals 1 + Sum_{k>=1} 2^k*Fibonacci(k-1)/k!.
Equals e * (cosh(sqrt(5)) - sinh(sqrt(5))/sqrt(5)). (End)

Extensions

a(100)-a(102) from Georg Fischer, Feb 08 2025

A193028 Decimal expansion of the coefficient of x in the reduction of e^(2x) by x^2->x+1.

Original entry on oeis.org

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

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

			11.244289366951194632995431729275126971...

Crossrefs

Cf. A000045, A193010, A192232, A193027.

Programs

  • Mathematica
    f[x_] := Exp[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^k*Fibonacci(k)/k!.
Equals 2*e*sinh(sqrt(5))/sqrt(5). (End)

Extensions

Offset corrected by Amiram Eldar, Jan 18 2022

A193029 Decimal expansion of the constant term of the reduction of e^(x/2) by x^2->x+1.

Original entry on oeis.org

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

			1.151945206759468821277485451505582743212385890...

Crossrefs

Cf. A000045, A193010, A192232, A193030.

Programs

  • Mathematica
    f[x_] := Exp[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 18 2022: (Start)
Equals 1 + Sum_{k>=1} Fibonacci(k-1)/(k!*2^k).
Equals exp(1/4)*(cosh(sqrt(5)/4) - sqrt(5)*sinh(sqrt(5)/4)/5). (End)
Previous Showing 11-20 of 35 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.