  We solve the equation of motion of Witten's cubic open string field theory in
a series expansion using the regulated butterfly state. The expansion parameter
is given by the regularization parameter of the butterfly state, which can be
taken to be arbitrarily small. Unlike the case of level truncation, the
equation of motion can be solved for an arbitrary component of the Fock space
up to a positive power of the expansion parameter. The energy density of the
solution is well-defined and remains finite even in the singular butterfly
limit, and it gives approximately 68% of the D25-brane tension for the solution
at the leading order. Moreover, it simultaneously solves the equation of motion
of vacuum string field theory, providing support for the conjecture at this
order. We further improve our ansatz by taking into account next-to-leading
terms, and find two numerical solutions which give approximately 88% and 109%,
respectively, of the D25-brane tension for the energy density. These values are
interestingly close to those by level truncation at level 2 without gauge
fixing studied by Rastelli and Zwiebach and by Ellwood and Taylor.
