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 20 results.

A299622 Decimal expansion of W(1) + W(1/2), where W is the Lambert W function (or PowerLog); see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Mar 03 2018

Keywords

Comments

The Lambert W function satisfies the functional equations
W(x) + W(y) = W(x*y*(1/W(x) + 1/W(y))) = log(x*y)/(W(x)*W(y)) for x and y greater than -1/e, so that W(1) + W(1/2) = W((1/2)*(1/W(1) + 1/W(1/2))) = -log(2) - log(W(1)) - log(W(1/2)). See A299613 for a guide to related sequences.

Examples

			W(1) + W(1/2) = 0.918877001658979699024877963140306614925280002...

Links

Crossrefs

Cf. A299613, A299623.

Programs

  • Mathematica
    w[x_] := ProductLog[x]; x = 1; y = 1/2; u = N[w[x] + w[y], 100]
RealDigits[u, 10][[1]]  (* A299622 *)
RealDigits[LambertW[1] + LambertW[1/2], 10, 100][[1]] (* G. C. Greubel, Mar 03 2018 *)
  • PARI
    lambertw(1) + lambertw(1/2) \\ G. C. Greubel, Mar 03 2018

A299623 Decimal expansion of e^(W(1) + W(1/2)) = (1/2)/(W(1)*W(1/2)), where W is the Lambert W function (or PowerLog); see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Mar 03 2018

Keywords

Comments

The Lambert W function satisfies the functional equation e^(W(x) + W(y)) = x*y/(W(x)*W(y)) for x and y greater than -1/e, so that e^(W(1) + W(1/2)) = (1/2)/(W(1)*W(1/2)). See A299613 for a guide to related constants.

Examples

			e^(W(1) + W(1/2)) = 2.506474042663898899474485815318941717...

Links

Crossrefs

Cf. A299613, A299622.

Programs

  • Mathematica
    w[x_] := ProductLog[x]; x = 1; y = 1/2;
N[E^(w[x] + w[y]), 130]   (* A299623 *)
RealDigits[1/(2*LambertW[1]*LambertW[1/2]), 10, 100][[1]] (* G. C. Greubel, Mar 03 2018 *)
  • PARI
    1/(2*lambertw(1)*lambertw(1/2)) \\ G. C. Greubel, Mar 03 2018

A299625 Decimal expansion of e^(2*W(2)) = 4/(W(2))^2, where W is the Lambert W function (or PowerLog); see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Mar 03 2018

Keywords

Comments

The Lambert W function satisfies the functional equation e^(W(x) + W(y)) = x*y/(W(x)*W(y)) for x and y greater than -1/e, so that e^(2*W(2)) = 4/(W(2))^2. See A299613 for a guide to related constants.

Examples

			e^(2*W(2)) = 5.50254660422072407531126813594932...

Links

Crossrefs

Cf. A299613, A299624.

Programs

  • Mathematica
    w[x_] := ProductLog[x]; x = 2; y = 2;
N[E^(w[x] + w[y]), 130]   (* A299625 *)
RealDigits[(2/LambertW[2])^2, 10, 100][[1]] (* G. C. Greubel, Mar 03 2018 *)
  • PARI
    (2/lambertw(2))^2 \\ G. C. Greubel, Mar 03 2018

A299626 Decimal expansion of 2*W(3), where W is the Lambert W function (or PowerLog); see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Mar 03 2018

Keywords

Comments

The Lambert W function satisfies the functional equations
W(x) + W(y) = W(x*y*(1/W(x) + 1/W(y))) = log(x*y)/(W(x)*W(y)) for x and y greater than -1/e, so that 2*W(3) = W(18/W(3)) = 2*(log(3) - log(W(3))). See A299613 for a guide to related sequences.

Examples

			2*W(3) = 2.0998177899280799199773941411...

Links

Crossrefs

Cf. A299613, A299627.

Programs

  • Mathematica
    w[x_] := ProductLog[x]; x = 3; y = 3; u = N[w[x] + w[y], 100]
RealDigits[u, 10][[1]]  (* A299626 *)
RealDigits[2*LambertW[3], 10,100][[1]] (* G. C. Greubel, Mar 03 2018 *)
  • PARI
    2*lambertw(3) \\ G. C. Greubel, Mar 03 2018

A299627 Decimal expansion of e^(2*W(3)) = 9/(W(3))^2, where W is the Lambert W function (or PowerLog); see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Mar 03 2018

Keywords

Comments

The Lambert W function satisfies the functional equation e^(W(x) + W(y)) = x*y/(W(x)*W(y)) for x and y greater than -1/e, so that e^(2*W(3)) = 9/(W(3))^2. See A299613 for a guide to related constants.

Examples

			e^(2*W(3)) = 8.1646820897128405910938873711...

Links

Crossrefs

Cf. A299613, A299626.

Programs

  • Mathematica
    w[x_] := ProductLog[x]; x = 3; y = 3;
N[E^(w[x] + w[y]), 130]   (* A299627 *)
RealDigits[(3/LambertW[3])^2, 10, 100][[1]] (* G. C. Greubel, Mar 06 2018 *)
  • PARI
    (3/lambertw(3))^2 \\ G. C. Greubel, Mar 06 2018

A299628 Decimal expansion of 2*W(1/3), where W is the Lambert W function (or PowerLog); see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Mar 03 2018

Keywords

Comments

The Lambert W function satisfies the functional equations
W(x) + W(y) = W(x*y*(1/W(x) + 1/W(y))) = log(x*y)/(W(x)*W(y)) for x and y greater than -1/e, so that 2*W(1/3) = W(2/(9*W(1/3))) = -2*log(3) - 2*log(W(1/3)). See A299613 for a guide to related sequences.

Examples

			2*W(1/3) = 0.5152553060994734085658324032521955...

Links

Crossrefs

Cf. A299613, A299629.

Programs

  • Mathematica
    w[x_] := ProductLog[x]; x = 1/3; y = 1/3; u = N[w[x] + w[y], 100]
RealDigits[u, 10][[1]]  (* A299628 *)
RealDigits[2*LambertW[1/3], 10, 100][[1]] (* G. C. Greubel, Mar 06 2018 *)
  • PARI
    2*lambertw(1/3) \\ G. C. Greubel, Mar 06 2018

A299629 Decimal expansion of e^(2*W(1/3)) = (1/9)/(W(1/3))^2, where W is the Lambert W function (or PowerLog); see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Mar 13 2018

Keywords

Comments

The Lambert W function satisfies the functional equation e^(W(x) + W(y)) = x*y/(W(x)*W(y)) for x and y greater than -1/e, so that e^(2*W(1/3)) = (1/9)/(W(1/3))^2. See A299613 for a guide to related constants.

Examples

			e^(2*W(1/3)) = 1.674065846464880772226081114...

Links

Crossrefs

Cf. A299613, A299628.

Programs

  • Mathematica
    w[x_] := ProductLog[x]; x = 1/3; y = 1/3; N[E^(w[x] + w[y]), 130]   (* A299629 *)
  • PARI
    exp(2*lambertw(1/3)) \\ Altug Alkan, Mar 13 2018

A299630 Decimal expansion of 2*W(3/2), where W is the Lambert W function (or PowerLog); see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Mar 13 2018

Keywords

Comments

The Lambert W function satisfies the functional equations
W(x) + W(y) = W(x*y*(1/W(x) + 1/W(y))) = log(x*y)/(W(x)*W(y)) for x and y greater than -1/e, so that 2*W(3/2) = W(9/(2*W(3/2))) = 2*log(3/2) - 2*log(W(3/2)). See A299613 for a guide to related sequences.

Examples

			2*W(3/2) = 1.451722715532452514097378798552...

Links

Crossrefs

Cf. A299613, A299631.

Programs

  • Mathematica
    w[x_] := ProductLog[x]; x = 3/2; y = 3/2; u = N[w[x] + w[y], 100]
RealDigits[u, 10][[1]]  (* A299630 *)
  • PARI
    2*lambertw(3/2) \\ Altug Alkan, Mar 13 2018

A299631 Decimal expansion of e^(2*W(3/2)) = (9/4)/(W(3/2))^2, where W is the Lambert W function (or PowerLog); see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Mar 13 2018

Keywords

Comments

The Lambert W function satisfies the functional equation e^(W(x) + W(y)) = x*y/(W(x)*W(y)) for x and y greater than -1/e, so that e^(2*W(3/2)) = (9/4)/(W(3/2))^2. See A299613 for a guide to related constants.

Examples

			e^(2*W(3/2)) = 4.2704649783213837050754449...

Links

Crossrefs

Cf. A299613, A299630.

Programs

  • Mathematica
    w[x_] := ProductLog[x]; x = 3/2; y = 3/2;
N[E^(w[x] + w[y]), 130]   (* A299631 *)
  • PARI
    exp(2*lambertw(3/2)) \\ Altug Alkan, Mar 13 2018

A299633 Decimal expansion of e^(2*W(e/2)) = (e^2/4)/(W(e/2))^2, where W is the Lambert W function (or PowerLog); see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Mar 13 2018

Keywords

Comments

The Lambert W function satisfies the functional equation e^(W(x) + W(y)) = x*y/(W(x)*W(y)) for x and y greater than -1/e, so that e^(2*W(e/2)) = (e^2/4)/(W(e/2))^2. See A299613 for a guide to related constants.

Examples

			e^(2*W(e/2)) = 3.9359563307913488100211988489777007...

Links

Crossrefs

Cf. A299613, A299632.

Programs

  • Mathematica
    w[x_] := ProductLog[x]; x = e/2; y = e/2; N[E^(w[x] + w[y]), 130]   (* A299633 *)
  • PARI
    exp(2*lambertw(exp(1)/2)) \\ Altug Alkan, Mar 13 2018
Previous Showing 11-20 of 20 results.

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