  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.
