This section aims to derive explicit mathematical expressions for computing the first Zagreb connection index and the second Zagreb connection index for a polyomino chain with n squares (P C n ) denoted as ZC1 n and ZC2 n , respectively. Let m ≥ 1 and i ∈ {1, 2, . . . , m} be given. It is worth noting that a polyomino chain P C n consists of a sequence of segments s 1 , s 2 , . . . , s m (refer to Figure ) with lengths l (s i ) = l i , satisfying the condition that m i=1 l i = n + m -1. Moreover, a chain of polyominoes P C n induces a sequence of 1s and 2s as follows: If we denote L n as a random variable with rank {1, 2} such that p = P(L n = 1), i.e., L n represents the selected link at time n, then we have a link of type j at time n if L n = j. Having established this information, we proceed to present the two fundamental results of this section. Proof. Given P C n , conducting the one-step analysis