[seqfan] (x+1)^(x+1)= x^(x+2) and x^x = (x-1)^(x+1)

Daniel Forgues kephalopod at gmail.com
Sat Jun 19 22:46:54 CEST 2010


[http://www.wolframalpha.com/input/?i=%28x%2B1%29^%28x%2B1%29+%3D+x^%28x%2B2%29
(x+1)^(x+1)= x^(x+2)]

gives:
 x~~3.141041525410788500945231446733515159979856852445599488196546631496424113176486717028008922619733816396791510643825934571540309860365903143378733054296284455377...

A144211 Decimal expansion of the convergent to x = 1/(x^(1/(x+1))-1) for x > 1.

which does not mention the equivalent form (x+1)^(x+1) = x^(x+2) which
make it obvious that it relates to (offset by 1):

A100086 Decimal expansion of unique positive solution to y^y=(y-1)^(y+1).

x~~4.141041525410788500945231446733515159979856852445599488196546631496424113176486717028008922619733816396791510643825934571540309860365903143378733054296284455377...

instead of:

Apparently related to A100086. [From R. J. Mathar
(mathar(AT)strw.leidenuniv.nl), Sep 17 2008]

Thanks

Daniel Forgues




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