Incremental rolling variance with no memory

Question

How can I compute a rolling variance WITHOUT keeping a buffer of input or other values?

In this old article:

https://www.dsprelated.com/showthread/comp.dsp/97276-1.php

I see that the mean required can be approximated with a exponential moving average, but I don't understand how I can then use that for calculating a variance?

In the article "John E. Hadstate" states:

A simpler way is to do this....

s(k+1)=beta*s(k)+(1-beta)*u^2(k)

where beta is a forgetting factor less than unity.

u is the input and s is the variance.

The correct equation for recursive variance can be found by addoing an extra sample to batch variance

s(k+1)=s(k)+(1/(k+1)[s(k)-u^2(k)]

But I don't understand that last line.

Does any one have a python version of this?