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Search: id:A141606
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| A141606 |
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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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4
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| 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
(list; cons; graph; listen)
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OFFSET
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1,2
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COMMENT
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Solution for x in x^(x^(e-1)) = e.
(W((y-1)ln(z))/((y-1)ln(z)))^(1/(1-y)) = e^(W((y-1)ln(z))/(y-1)) so that (W(e-1)/(e-1))^(1/(1-e)) = e^(W(e-1)/(e-1)). [From Ross La Haye (rlahaye(AT)new.rr.com), 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 this way: (x^x)^x)^x = x^(x^3). In general, this type of bracketing will simplify to x^(x^(y-1)) when y xs appear in the expression. Solving the equation x^(x^(y-1)) = z for x gives x = (W((y-1)ln(z))/((y-1)ln(z)))^(1/(1-y)). And setting y = z = e gives the result indicated by this sequence. I thank Mike Wentz with introducing me to the expression x^(x^(y-1)) that results from the "bottom up" bracketing of x^x^x^x... and the motivation for its investigation.
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LINKS
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Eric Weisstein, Lambert W-Function
Eric Weisstein, Power Tower
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EXAMPLE
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1.57844691419127618691147145725058871862508588172697263709178296257...
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MATHEMATICA
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RealDigits[N[(ProductLog[E-1]/(E-1))^(1/(1-E)), 111][[1]]
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CROSSREFS
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Cf. A001113.
Cf. A143913, A143914, A143915. [From Ross La Haye (rlahaye(AT)new.rr.com), Sep 05 2008]
Sequence in context: A153104 A155855 A070366 this_sequence A068001 A068213 A065966
Adjacent sequences: A141603 A141604 A141605 this_sequence A141607 A141608 A141609
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KEYWORD
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cons,nonn
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AUTHOR
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Ross La Haye (rlahaye(AT)new.rr.com), Aug 21 2008, Aug 26 2008
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EXTENSIONS
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More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 25 2008
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