Alternating sequences

AbstractLet a=(α1, α2, α3, …) be a sequence of positive integers. The sequence (c1, c2, …, c3) is a-alternating if 1 ⩽ c1 < c2 < … < ck ⩽ n and in additionthe first α1 elements have the same parity, the next α2 elements have opposite parity, the next α3 elements have the parity of the first group, and so on. The final group of elements of like parity is permitted to have fewer elements than the required number. Let ƒ(a; n, k) denote the number of a-alternating sequences of length k. An explicit formula for ƒ (a;n, k) is obtained