Find Number of Ways to Reach the K-th Stair

Count the number of operation sequences starting from stair 1 with jump=0 that can reach stair k, where each step is either down by 1 (not allowed from stair 0 or twice consecutively) or up by 2^jump (after which jump increments). This is a counting/DP problem over states (position, jump, last-move) requiring careful handling of growing power-of-two jumps and revisits.