Jacobi's cubic analog of the pentagonal number theorem and representations of 24n+5 as a sum of two squares

In this paper, we consider the two squares problem in order to introduce a combinatorial interpretation for Jacobi's cubic analog of Euler's pentagonal number theorem. Under certain conditions imposed by Fermat's theorem on representations of integers as a sum of two squares, we derive a linear homogeneous recurrence relation for Euler's partition function. In this context, we introduce two infinite conjectural families of Ramanujan type congruences and prove several special cases.