Python 2, 76 bytes
i=27
while~i:print' '*i+' '.join(`min(i,j,27-i-j)`for j in range(28-i));i-=1
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Computes the digit in row i from the bottom, position j as min(i,j,27-i-j).
F...
Haskell, 20 bytes
f x=foldl((+).(x*))0
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Takes input coefficients from highest degree to lowest.
21 bytes
x%(h:t)=h+x*x%t
x%_=0
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Python 3, 35 bytes
f=lambda x,i=1:x>0and-~f(x-1/i,i+1)
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Subtracts each 1/i in turn from the initial value until the result is no longer positive.
Python 2, 37 bytes
l=input()
while 1:print len(l);l=l[0]
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This makes heavy use of programs being allowed to terminate with error, which I assume is allowed by default because ...
