Conjectured recursive relationship for expected values in the coin-toss automaton

Prove that in the finite automaton modeling the symmetric coin-toss process used to analyze the expected number of transactions in a grid trading strategy—where q_i denotes the state with head–tail difference i and E_i denotes the expected number of steps starting from state q_i—the expected values satisfy the recursive relationship E_m = E_0 − m^2 for all integers m ≥ 1.

Background

To compute the expected number of transactions for a grid trading strategy under the assumption that price moves up or down by a fixed ratio k with equal probability, the authors model the problem as a symmetric coin-toss process. In this model, states q_i represent the difference between the number of heads and tails, and E_i denotes the expected number of steps remaining from state q_i to reach the terminal boundary state.

From observed relationships among E_i (e.g., E_i = 2E_{i-1} − E_{i-2} − 2), the authors posit a closed-form recursive relationship for E_m in terms of E_0 and the index m, namely E_m = E_0 − m^2. They explicitly frame this as a conjecture before presenting an induction-based derivation.