It looks like MSRI is getting into serious outreach.
Bob Baillie passed along the url
http://www.numberphile.com
I haven't looked at the insides yet, but the front page
looks tasty.
Rich
Define A(x) by A(x) = 1 + x / A(x^2), so
A(x) = 1+x/(1+x^2/(1+x^4/(1+x^8/(1+ ...)))).
Define F(x) by F(x) = 1 + x / (1 - x^2 / F(x^2) ), so
F(x) = 1+x/ (1-x^2/ (1+x^2/ (1-x^4/ (1+x^4 /(1-x^8/ (1+x^8/ (1-x^16/ ... ))))))).
The following took me a while:
Show that F(x) - x = A(x^3)
Spoiler below. I am asking because I'd like to know how
you'd rate it in difficulty and time this should/did take.
Regards, jj
... on to the spoiler
... on to the spoiler
... on to the spoiler
******************** SPOILER **********************
Set G(x) = F(x) - x
observe that
G(x) = 1 + x^3/G(x^2)
= 1 + x^3/(1 + x^6 / G(x^2) ) = ...
= 1 + x^3/(1 + x^6 / (1 + x^12 / (1 + x^24 / (...) ) ) )
= A(x^3).
Cf.
https://oeis.org/draft/A238429 (expansion of F),
https://oeis.org/draft/A218031 (expansion of A)
