Patterns in Continued Fractions of Square Roots

We examine the structure of the periodic continued fractions of square roots of non-square positive integers given by an integer-valued quadratic polynomial Q(n) = (a(n) + b)(2) + (gn + h). The aim is to identify repeated blocks of partial quotients in the period. The quotients in the period form a palindrome, and when the period length is even, the period has a central term an. The paper focuses on periods with a(n) = a(0) or a(n) = a(0 - 1), where a(0) is the initial partial quotient. For a(n) = a(0) we give an algorithm to obtain formulas involving repeated blocks comprising three or more elements, not all equal.