We study a family of symmetric third-order recurring sequences with the aid of Riordan arrays and Chebyshev polynomials. Formulas involving both Chebyshev polynomials and Fibonacci numbers are established. The family of sequences defined by the product of consecutive terms of the first family of sequences is also studied, and links to the Chebyshev polynomials are again established, including continued fraction expressions. A multiplicative result is established relating Chebyshev polynomials to sequences of doubled Chebyshev polynomials. Links to a special Catalan related Riordan array are explored.