  The main step in numerical evaluation of classical Sl2 (Z) modular forms and
elliptic functions is to compute the sum of the first N nonzero terms in the
sparse q-series belonging to the Dedekind eta function or the Jacobi theta
constants. We construct short addition sequences to perform this task using N +
o(N) multiplications. Our constructions rely on the representability of
specific quadratic progressions of integers as sums of smaller numbers of the
same kind. For example, we show that every generalised pentagonal number c 5
can be written as c = 2a + b where a, b are smaller generalised pentagonal
numbers. We also give a baby-step giant-step algorithm that uses O(N/ log r N)
multiplications for any r > 0, beating the lower bound of N multiplications
required when computing the terms explicitly. These results lead to speed-ups
in practice.
