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.

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A193030 Decimal expansion of the coefficient of x in the reduction of e^(x/2) by x^2->x+1.

Original entry on oeis.org

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

			0.675977245872051077662259637417563070...

Crossrefs

Cf. A000045, A193010, A192232, A193029.

Programs

  • Mathematica
    f[x_] := Exp[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 18 2022: (Start)
Equals Sum_{k>=1} Fibonacci(k)/(k!*2^k).
Equals 2*exp(1/4)*sinh(sqrt(5)/4)/sqrt(5). (End)

A193031 Decimal expansion of the constant term of the reduction of 2^x by x^2->x+1.

Original entry on oeis.org

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

			1.3198787240211562744599742126393139318590...

Crossrefs

Cf. A000045, A001622, A193010, A192232, A193032.

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 (1 + (3+4^phi)/sqrt(5))/(phi*2^phi), where phi is the golden ratio (A001622). (End)

A193032 Decimal expansion of the coefficient of x in the reduction of 2^x by x^2->x+1.

Original entry on oeis.org

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

			1.081365283916960767554821105448442602497...

Crossrefs

Cf. A000045, A001622, A193010, A192232, A193031.

Programs

  • Mathematica
    f[x_] := 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 19 2022: (Start)
Equals Sum_{k>=0} log(2)^k*Fibonacci(k)/k!.
Equals (2^sqrt(5)-1)/(sqrt(5)*2^(phi-1)), where phi is the golden ratio (A001622). (End)

A193035 Decimal expansion of the coefficient of x in the reduction of 2^(-x) by x^2->x+1.

Original entry on oeis.org

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

			-0.540682641958480383777410552724221301248532691111...

Crossrefs

Cf. A000045, A001622, A193010, A192232, A193009.

Programs

  • Mathematica
    f[x_] := 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 19 2022: (Start)
Equals Sum_{k>=0} (-log(2))^k*Fibonacci(k)/k!.
Equals -(2^sqrt(5) - 1)/(sqrt(5)*2^phi), where phi is the golden ratio (A001622). (End)

Extensions

a(99) corrected by Georg Fischer, Aug 04 2024

A193075 Decimal expansion of the constant term of the reduction of phi^x by x^2->x+1, where phi=(1+sqrt(5))/2 is the golden ratio (A001622).

Original entry on oeis.org

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

			1.13956470687932160823788165057931871131735800...

Crossrefs

Cf. A000045, A001622, A193010, A192232, A193076.

Programs

  • Mathematica
    t = GoldenRatio
f[x_] := t^(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(phi)^k*Fibonacci(k-1)/k!.
Equals (sqrt(5)*phi^sqrt(5) + phi^4 - 1)/(5*phi^phi). (End)

A193076 Decimal expansion of the coefficient of x in the reduction of phi^x by x^2->x+1, where phi=(1+sqrt(5))/2 is the golden ratio (A001622).

Original entry on oeis.org

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

			0.6420710988036375722663484493183969433220...

Crossrefs

Cf. A000045, A001622, A193010, A192232, A193075.

Programs

  • Mathematica
    t = GoldenRatio
f[x_] := t^(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(phi)^k*Fibonacci(k)/k!.
Equals (phi^phi - phi^(1-phi))/sqrt(5). (End)

Extensions

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

A193077 Decimal expansion of the constant term of the reduction of phi^(-x) by x^2->x+1, where phi=(1+sqrt(5))/2 is the golden ratio (A001622).

Original entry on oeis.org

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

			1.101111483485871838026720619840...

Crossrefs

Cf. A000045, A001622, A193010, A192232, A193078.

Programs

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

A193078 Decimal expansion of the coefficient of x in the reduction of phi^(-x) by x^2->x+1, where phi=(1+sqrt(5))/2 is the golden ratio (A001622) (negated).

Original entry on oeis.org

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

			-0.39682176225463996686831560297353019716760...

Crossrefs

Cf. A000045, A001622, A193010, A192232, A193077.

Programs

  • Mathematica
    t = GoldenRatio
f[x_] := t^(-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(phi))^k*Fibonacci(k)/k!.
Equals -(phi^sqrt(5) - 1)/(sqrt(5)*phi^phi). (End)

A193079 Decimal expansion of the constant term of the reduction of sinh(2x) by x^2->x+1.

Original entry on oeis.org

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

			2.3691586979000944404682728022583852586980...

Crossrefs

Cf. A000045, A193010, A192232, A193080.

Programs

  • Mathematica
    f[x_] := Sinh[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 Sum_{k>=1} 2^(2*k+1)*Fibonacci(2*k)/(2*k+1)!.
Equals ((3+sqrt(5))*sinh(1-sqrt(5)) + 2*sinh(1+sqrt(5)))/(5 + sqrt(5)). (End)

A193080 Decimal expansion of the coefficient of x in the reduction of sinh(2x) by x^2->x+1.

Original entry on oeis.org

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

			6.3830192266109834906946736316102032592390...

Crossrefs

Cf. A000045, A193010, A192232, A193079.

Programs

  • Mathematica
    f[x_] := Sinh[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>=0} 2^(2*k+1)*Fibonacci(2*k+1)/(2*k+1)!.
Equals 2*cosh(1)*sinh(sqrt(5))/sqrt(5). (End)
Previous Showing 21-30 of 35 results. Next

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