A143915 -id:A143915 - cp's OEIS frontend

A141606 Decimal expansion of (W(e-1)/(e-1))^(1/(1-e)), where W(z) denotes the Lambert W function and e = 2.718281828...

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

Offset: 1

Ross La Haye, Aug 21 2008, Aug 26 2008

Solution for x in x^(x^(e-1)) = e.
(W((y-1)*log(z))/((y-1)*log(z)))^(1/(1-y)) = e^(W((y-1)*log(z))/(y-1)) so that (W(e-1)/(e-1))^(1/(1-e)) = e^(W(e-1)/(e-1)). - Ross La Haye, Aug 27 2008
Consider the expression x^x^x^x... where x appears y times. For, say, y = 4 this type of expression is conventionally evaluated as if bracketed x^(x^(x^x)) and is referred to as a "power tower". However, we can also bracket x^x^x^x from the bottom up, e.g., ((x^x)^x)^x = x^(x^3). In general, this bracketing will simplify x^x^x^x... to x^(x^(y-1)) when x appears y times in the expression. Solving the equation x^(x^(y-1)) = z for x gives x = (W((y-1)*log(z))/((y-1)*log(z)))^(1/(1-y)). And setting y = z = e gives the result indicated by this sequence. Special thanks are due to Mike Wentz for introducing me to the "bottom up" bracketing of x^x^x^x... and the motivation for its investigation.

			1.57844691419127618691147145725058871862508588172697263709178296257...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein, Lambert W-Function
Eric Weisstein, Power Tower

Cf. A001113.

Cf. A143913, A143914, A143915. - Ross La Haye, Sep 05 2008

RealDigits[(ProductLog[E-1]/(E-1))^(1/(1-E)),10,111][[1]]

(lambertw(exp(1)-1)/(exp(1)-1))^(1/(1-exp(1))) \\ G. C. Greubel, Mar 02 2018

More terms from Robert G. Wilson v, Aug 25 2008

A143913 Continued fraction expansion of e^(W(e-1)/(e-1)) where W(z) denotes the Lambert W function and e = 2.718281828...

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 5, 5, 1, 554, 3, 4, 1, 23, 2, 9, 2, 1, 305, 1, 1, 1, 3, 5, 5, 1, 3, 10, 5, 3, 1, 2, 1, 2, 1, 1, 2, 1, 4, 2, 1, 3, 2, 9, 1, 4, 11, 1, 3, 1, 3, 3, 1, 11, 4, 4, 1, 1, 2, 2, 3, 3, 4, 1, 3, 6, 3, 1, 2, 1, 3, 3, 3, 9, 1, 2, 11, 7, 1, 4, 1, 8, 10, 23, 2, 4, 1, 2, 10, 5, 2, 5, 1, 3, 36, 1, 6, 1, 6

Offset: 0

Ross La Haye, Sep 04 2008

G. C. Greubel, Table of n, a(n) for n = 0..9999

Cf. A141606 for decimal expansion.

Cf. A143914, A143915.

ContinuedFraction[E^((ProductLog[E - 1])/(E - 1)), 111]

contfrac(exp((lambertw(exp(1) -1)/(exp(1) -1)))) \\ G. C. Greubel, Mar 02 2018

Offset changed by Andrew Howroyd, Aug 09 2024

A143914 Numerators of continued fraction convergents to e^(W(e-1)/(e-1)) where W(z) denotes the Lambert W function and e = 2.718281828...

Original entry on oeis.org

1, 2, 3, 8, 11, 30, 161, 835, 996, 552619, 1658853, 7188031, 8846884, 210666363, 430179610, 4082282853, 8594745316, 12677028169, 3875088336861, 3887765365030, 7762853701891, 11650619066921, 42714710902654

Offset: 0

Ross La Haye, Sep 05 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A141606 for decimal expansion, A143913 for continued fraction expansion.

Cf. A143915

Table[Numerator[FromContinuedFraction[ContinuedFraction[E^((ProductLog[E - 1])/(E - 1)), n]]], {n, 1, 25}]

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A141606 Decimal expansion of (W(e-1)/(e-1))^(1/(1-e)), where W(z) denotes the Lambert W function and e = 2.718281828...

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A143913 Continued fraction expansion of e^(W(e-1)/(e-1)) where W(z) denotes the Lambert W function and e = 2.718281828...

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A143914 Numerators of continued fraction convergents to e^(W(e-1)/(e-1)) where W(z) denotes the Lambert W function and e = 2.718281828...

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