Paperfolding, Automata, and Rational Functions - Diagonals and ...

An Algebraic Equation
The function S(X) = P ∞
h=0 shX h therefore satisfies
(1 − X 2 )(1 − X)S(X 2 ) − (1 − X 2 )S(X) + X = 0 .
Exercise. Show that for an arbitrary sequence (ih), with ih ∈ {0, 1},
one has P (−1) i hX h = (1 − X) −1 − 2 P ihX h , and hence confirm the
“therefore” above.
Hence, reducing modulo 2, we see that over the finite field F2 we have
(1 + X) 3 S 2 + (1 + X) 2 S + X = 0 ,
showing that S is quadratic irrational over the field F2(X).
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