Horadam introduced a new generalized sequence of numbers, describing its key features and the special sub-sequences that are obtained depending on the choices of initial parameters. This sequence and its sub-sequences are known as the Horadam, generalized Fibonacci, and generalized Lucas numbers, respectively. In the present study, we propose another new sequence, which satisfies a second-order recurrence relation, in addition to Horadam's definition. Further, we prove the Binet's formula, some famous identities, and summation formulas for this new sequence. In particular, we demonstrate the interrelationships between our new sequence and the Horadam sequence.