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 1-10 of 35 results. Next

A099935 Decimal expansion of Sum_{k>=0} (-1)^(k+1)*A000045(k)/k!.

Original entry on oeis.org

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

Views

Author

Benoit Cloitre, Nov 12 2004

Keywords

Comments

This number is the coefficient of x in the reduction of e^(-x) by the substitution x^2->x+1; see A193026 and A193010.

Examples

			0.74102792152357735584178398667102441173255884250150...

Crossrefs

Cf. A000045, A001113, A098689, A193010, A193026.

Programs

  • Mathematica
    (E^(1+2/GoldenRatio)-1) / (E^GoldenRatio*(2*GoldenRatio-1)) // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Feb 13 2013 *)
  • PARI
    exp(-1/2)*(2/sqrt(5))*sinh(sqrt(5)/2) \\ Michel Marcus, Feb 07 2022

Formula

Equals exp(-1/2)*(2/sqrt(5))*sinh(sqrt(5)/2).
Equals A098689 / e. - Amiram Eldar, Feb 07 2022

A193009 Decimal expansion of the constant term of the reduction of 2^(-x) by x^2->x+1.

Original entry on oeis.org

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

			1.20062200396905852100739765904387826717805500...

Crossrefs

Cf. A000045, A001622, A193010, A192232, A193035.

Programs

  • Mathematica
    f[x_] := 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 19 2022: (Start)
Equals 1 + Sum_{k>=1} (-log(2))^k*Fibonacci(k-1)/k!.
Equals (4^(phi-1)*(5+3*sqrt(5)) + sqrt(5))/(5*phi*2^phi), where phi is the golden ratio (A001622). (End)

A193014 Decimal expansion of the coefficient of x in the reduction of cos(x) by x^2->x+1.

Original entry on oeis.org

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

			-0.3856051379557488291513243591721325...

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]
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} (-1)^k*Fibonacci(2*k)/(2*k)!.
Equals -2*sin(1/2)*sin(sqrt(5)/2)/sqrt(5). (End)

A193081 Decimal expansion of the constant term of the reduction of cosh(2x) by x^2->x+1.

Original entry on oeis.org

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

			4.8707186334008057368566594211377757838921196937...

Crossrefs

Cf. A000045, A193010, A192232, A193082.

Programs

  • Mathematica
    f[x_] := Cosh[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^(2*k)*Fibonacci(2*k-1)/(2*k)!.
Equals ((3+sqrt(5))*cosh(1-sqrt(5)) + 2*cosh(1+sqrt(5)))/(5 + sqrt(5)). (End)

A193083 Decimal expansion of the constant term of the reduction of (e^x)*cos(x) by x^2->x+1.

Original entry on oeis.org

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

			constant=0.25205898406677536274258477825247829436590...

Crossrefs

Cf. A193010, A192232, A193084.

Programs

  • Mathematica
    f[x_] := Exp[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, 100}], 100]
RealDigits[u0, 10]  (* A193083 *)
u1 = N[Sum[c[n]*r[n], {n, 0, 100}], 100]
RealDigits[u1, 10]  (* A193084 *)

A193084 Decimal expansion of the coefficient of x in the reduction of (e^x)*cos(x) by x^2->x+1.

Original entry on oeis.org

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

			-0.3029588714336260011975849802354811365293918921...

Crossrefs

Cf. A193010, A192232, A193083.

Programs

  • Mathematica
    f[x_] := Exp[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, 100}], 100]
RealDigits[u0, 10]  (* A193083 *)
u1 = N[Sum[c[n]*r[n], {n, 0, 100}], 100]
RealDigits[u1, 10]  (* A193084 *)

A193085 Decimal expansion of the constant term of the reduction of (e^x)*sin(x) by x^2->x+1.

Original entry on oeis.org

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

			constant=1.166347348190864540481321655470201723
...

Crossrefs

Cf. A193010, A192232, A193086.

Programs

  • Mathematica
    f[x_] := Exp[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, 100}], 100]
RealDigits[u0, 10]  (* A193085 *)
u1 = N[Sum[c[n]*r[n], {n, 0, 100}], 100]
RealDigits[u1, 10]  (* A193086 *)

A193086 Decimal expansion of the coefficient of x in the reduction of (e^x)*sin(x) by x^2->x+1.

Original entry on oeis.org

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

			2.3925286592724290973874921165892277198499...

Crossrefs

Cf. A193010, A192232, A193085.

Programs

  • Mathematica
    f[x_] := Exp[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, 100}], 100]
RealDigits[u0, 10]  (* A193085 *)
u1 = N[Sum[c[n]*r[n], {n, 0, 100}], 100]
RealDigits[u1, 10]  (* A193086 *)

A193087 Decimal expansion of the constant term of the reduction of (cos(x))^2 by x^2->x+1.

Original entry on oeis.org

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
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

			0.48127666131507080300697222296113439695...

Crossrefs

Cf. A000045, A001622, A193010, A192232, A193088, A193089.

Programs

  • 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]

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)

A193088 Decimal expansion of the coefficient of x in the reduction of (cos(x))^2 by x^2->x+1 (negated).

Original entry on oeis.org

2, 9, 6, 0, 6, 7, 2, 8, 1, 1, 2, 9, 8, 0, 8, 0, 5, 1, 0, 7, 9, 7, 3, 9, 7, 7, 0, 1, 3, 9, 6, 9, 6, 3, 9, 2, 9, 1, 6, 5, 2, 7, 5, 6, 4, 7, 6, 9, 2, 8, 5, 8, 3, 3, 3, 5, 0, 7, 5, 8, 0, 7, 9, 8, 7, 6, 2, 8, 9, 2, 5, 3, 0, 4, 5, 2, 5, 8, 2, 4, 8, 8, 3, 8, 4, 4, 3, 4, 7, 1, 8, 5, 1, 6, 5, 6, 5, 7, 3, 3, 9
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.
The positive value +0.2960... is the coefficient of x in the reduction of (sin(x))^2 by x^2->x+1.

Examples

			-0.29606728112980805107973977013969639291652...

Crossrefs

Cf. A000045, A193010, A192232, A193087.

Programs

  • Mathematica
    f[x_] := Cos[x]^2; 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} (-1)^k * 2^(2*k-1) * Fibonacci(2*k)/(2*k)!.
Equals -sin(1)*sin(sqrt(5))/sqrt(5). (End)

Extensions

a(99)-a(100) from Georg Fischer, Feb 08 2025
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