Expected flips for arbitrary ending strings

Determine whether, for every binary ending string S other than a constant string consisting solely of heads or solely of tails, with total length s, the expected number E(S) of coin flips until the first occurrence of S equals 2^s possibly plus some lower positive powers of 2, and consequently is asymptotic to 2^s.

Background

The paper studies the expected number E(S) of fair-coin flips required for a specified finite string S of heads and tails to occur for the first time. The authors derive exact formulas when S has at most four maximal runs or alternates between heads and tails; in all of these cases, the resulting expectations are sums involving powers of 2.

Based on these results, the authors conjecture a general form for E(S) for arbitrary ending strings, excluding strings consisting entirely of one symbol. The proposed form predicts that the dominant term depends only on the total length of S, not on its internal arrangement, and is intended to explain the observed power-of-two structure.