Euler's constant e can be expressed very prettily as an infinite continued fraction, following the pattern [1 0 1 1 2 1 1 4 1 1 6 1 1 8 1] where the (3n+2)th term is 2n and all other terms equal 1. This describes the following fraction:
1 + 1
_______
0 + 1
_______
1 + 1
_______
1 + 1
_______
2 + 1
_______
1 + 1
_______
1 + 1
_______
4 + 1
_______
1 + 1
_______
1 + 1
_______
6 + ...
Euler was probably the first to discover this (he used the expansion to prove
e's
irrationality, as well as that of its square). The zero up the top is of course somewhat redundant, however the alternative is to express the fraction as [2 1 2 1 1 4 1 1 6 1 1 8 1], which is not quite so homogenous a pattern and somewhat conceals the identity's
beauty. (This "zero trick" was invented by
Bill Gosper to smooth out what he saw as a glitch in the original representation.)